Eigil Fjeldgren Rischel 2025 4 6 https://erischel.com/efr-HQ73/ efr-HQ73 /efr-HQ73/ Markov Fibrations › Introduction In applying category theory to fields as diverse as game theory (Reference ,Reference , Reference ), machine learning (Reference , Reference ), dynamical systems (Reference , Reference ), and server design (Reference ), people have found it useful to study categories whose morphisms describe processes or functions that take place in two "stages", where the second is dependent on the first, but composes in the other direction---so-called lenses. In many of these domains, stochastic phenomena play a role. There is a useful generalization of lenses to categories of stochastic maps---the so-called optics of Riley, Reference ---but this does not accommodate another useful generalization, so called dependent lenses where not only the backwards process but the set it takes values in is indexed over the base. It has been a long-standing problem to develop a suitable common generalization, "dependent optics", of these two ideas. In this thesis, we solve this problem by developing a theory of Markov fibrations (§ ). A Markov fibration is a weakening of the notion of (Grothendieck) fibration to include (subject to some assumptions) Markov categories of indexed families of objects (given by deterministic functions ) and compatible stochastic maps (given by commutative squares). The main point of Markov fibrations is that they admit fiberwise opposites, which generalize the fiberwise opposites of ordinary categories. Just as the fiberwise opposite of the codomain fibration of a finitely complete category describes dependent lenses (see § ), the fiberwise opposite of these codomain Markov fibrations give a good notion of stochastic lens (see Theorem ). We will apply our theory chiefly to two problems. First, we will use them to generalize open games (Hedges, Reference ) to a larger class of interfaces (namely, indexed families of sets) while at the same time considering possibly-stochastic maps. This will allow us to construct the so-called external choice operator on open games, which describes branching. Secondly, we will generalize Myers' categorical dynamical systems theory to allow for stochastic maps in the base. When combined with another generalization of these systems, to general parametrized lenses, this gives a more natural way of modeling certain stochastic dynamical systems, such as those associated to the training dynamics of machine learning models, see eg Example . Eigil Fjeldgren Rischel 2025 4 6 A brief introduction to lenses If is a category with finite products, the category of lenses in has objects pairs of objects in , and morphisms given by pairs . (Note that it is of course the morphisms, not the objects, that are called lenses). One way to generalize this is to ask for to have pullbacks, and consider an object given by a more general map , and let a map be (so that the triangle over commutes). (Note that this recovers the "simple lenses" when the objects are of the form ). Under the interpretation of a map as a family of objects indexed over the elements of , we see this as a "dependent lens" (since is a "dependent type"). There are various variations of this idea, which ultimately all fit the pattern of taking a Grothendieck fibration and forming the fiberwise opposite---for the dependent lenses as above, this is the codomain fibration . Meanwhile, another wide-ranging generalization of are the optics of Riley (Reference ). For a monoidal category, the objects of are again pairs of objects, but now the morphisms are elements of the coend That is, to give an optic is to give an object and morphisms , up to the equivalence relation generated by, whenever identifying the two optics given by and . One can show that, in the case where is a Cartesian monoidal category, this set can be identified with the set of (simple) lenses. A vector field on a smooth manifold is simply a (smooth) section of the tangent bundle . A section is a bundle map from the trivial bundle . This gives the idea of considering a parametrized dynamical system as consisting of a map to some other smooth manifold, a bundle (the points of are the parameters) and a bundle map . As we will see, such a bundle map is exactly a dependent lens in a category of bundles. Building on this idea, Myers (Reference ) described a highly abstract theory of open dynamical systems, given by lenses out of tangent bundles (where the notion of space, bundle and tangent bundle are generalized to any fibration). It has been observed that, although a lens from the tangent bundle may indeed be said to describe an open dynamical system with state space , the same can be said for a lens of type ---although a dynamical system of a different kind (essentially, the first type are the Moore machines, the second the Mealy machines---see the introduction to § for more on this). It is natural to consider parametrized maps as a common generalization of these two concepts---and in fact, special cases of this idea, lenses parametrized by tangent bundles, have already been considered many times, for example in the semantics of gradient descent (see Reference , Reference , Reference ). Since parametrized maps compose in an obvious way, we now have three different notions of morphisms between bundles---lenses, charts, and "generalized systems". These should form some sort of symmetric monoidal triple category, but the right axiomatisation of this concept is somewhat elusive. Lenses, in their various incarnations, have been widely used in applied category theory. For example, Hedges and his collaborators have developed a compositional approach to game theory (Reference , Reference , Reference , Reference , others). Recent work by Hedges and Sakamoto bring this idea to reinforcement learning (Reference ). We briefly mentioned Myers' categorical systems theory above, and we will see much more of it later. As we discussed above, there is also a literature using lenses to study gradient descent in an abstract sense. This is without even mentioning their original role in the theory of functional programming (this is the origin of the somewhat confusing term lens), or their prehistory in Gödel's dialectica interpretation---see eg Reference for a survey of this. Eigil Fjeldgren Rischel 2025 4 6 Dependent Optics An open game, in the sense of Hedges, has as its interface two objects in the category of lenses (of sets) (that is, two pairs of sets). An open game with this interface consists of a set of strategies equipped with a function and a subset of equilibrium strategies for each and ---note that such an is exactly a lens and such a is exactly a lens . This view of open games makes the composition rule much simpler to define, although we will not delve into the details here, see eg Reference . The idea is that the open game represents the strategies and preferences of a player or set of players--- are the possible strategies, is the information revealed to the player before they make their decision of which moves to make, the resulting is the choice the player "sends" in response to , and the is the "utility", the value they eventually learn, depending on their choice , which they have some preferences over. The above description gives a theory of deterministic games, but of course, it is completely essential for game theory to model both random decisions (mixed strategies) and decisions made under uncertainty. This motivates the replacement of lenses in this definition with optics in a category of probability kernels (the form of the equilibrium relation must be modified as well, see Reference , Reference ) In the abstract study of open games, it is desirable to define a "choice" operation on the objects (the pairs of sets) so that maps into represent players who are faced with some binary choice between and , after which play may proceed for a time in one of two branches. However, this is immediately problematic because the category of lenses do not have coproducts---inside the category of dependent lenses, however, we can form the coproduct simply as ---that is, we get a family which is over , and over . This indeed does the job, but for game theory the extension to stochastic maps---optics---is absolutely essential. This motivates the search for a theory of dependent optics, a common generalization of optics and dependent lenses. There have been a number of attempts at this---see eg Reference , Reference , Reference , Reference . While these efforts have to some extent succeeded in describing a theory that is general enough to contain the desired examples, it is generally very ad hoc. The construction of optics as a special case of Vertechi's dependent optics (Reference ) gives them as fibre optics with the indexing bicategory being (the delooping of) a monoidal category , but embeds lenses using a bicategory of spans. Thus simple lenses admit two distinct encodings in this theory. Moreover, Vertechi's example of dependent monoidal optics fails to describe a supercategory of when applied to Markov categories, and are thus not suitable for, for example, game theory. Eigil Fjeldgren Rischel 2025 4 6 Categorical Cybernetics The observation that both open games and machine learning systems can be fruitfully described using lenses, naturally led to the idea that this could be used to develop a more thorough analogy between the two. After all, a machine learning system is also a player in a kind of game (with the objective of minimizing loss). Hedges coined the phrase categorical cybernetics to describe this locus of ideas (Reference , also used in Reference ). Myers' categorical systems theory also describes dynamical systems as lenses. However, where a cybernetic system in the sense of Reference is a parametrized lens and thus has two different inputs - which is the information on which basis they are allowed to choose their decision and which is their "utility", the information they are allowed to care about. It is straightforward to describe the machine learning part of the categorical cybernetics analogy in terms of Myers' theory---a "learner" is simply a lens of the above form with replace with a tangent bundle . This idea (described in these terms, although not with the analogy to machine learning) has already been described by Reference . However, for game theory, it is necessary that the output of a given system can be stochastically chosen (a player must be able to randomize their strategy). At the same time, for a machine learning system, the is usually some sort of sample from a training distribution---ie, it is random. Hence to really describe the training dynamics of machine learning systems using this theory, we are again naturally drawn to consider systems theories (that is, fibrations) which have stochastic maps in the base, not merely the fiber. Eigil Fjeldgren Rischel 2025 4 6 Overview and contributions We begin the thesis in § with a review of some preliminary material. This chapter can largely be skipped for readers already familiar with the material (Markov categories, double categories, lenses and optics, fibrations, and categorical dynamical systems theory). The chapter on Markov categories introduces a few novel auxiliary notions, but these can be referred back to as necessary (they are primarily regularity conditions which hold in most Markov categories of interest). In § , we give the main technical contribution of the thesis by developing a theory of Markov fibrations. Their fiberwise opposites generalize both the fiberwise opposites of codomain fibrations (dependent lenses in a classical sense) and optics in Markov categories. Theorem provides the statement of the latter (the former is straightforward). We also give a description of monoidal structures on Markov fibrations (which pass to monoidal structures on the resulting categories of optics). In § , we give a construction of the double category of parametrized morphisms for a category action (or actegory)---in fact, we do this internally to any -category. This has the advantage of allowing us to construct more highly-structured versions of this double category by carrying out the construction internally to higher-structured actions. (Bicategorical versions of are a relatively old idea, and the double categorical version is a straightforward extension which has existed for some time as folklore, but the fully-internal construction here is novel). In § , we apply Markov fibrations to compositional game theory, and solve a longstanding problem by giving a category of stochastic open games with a general "external choice" operator. This construction is largely abstract over the particular category, and so can be applied to different Markov fibrations to describe games defined with different notions of stochasticity. We also review some previous work with Capucci, Hedges, and Gavranovic on abstract constructions of categories of open games (Reference ). In § , we describe how to extend Myers' categorical systems theory to Markov fibrations, to give theories of systems which may have stochastic maps not merely in the fiber (as was already described by Myers in Reference ) but also in the base. We also leverage our construction from § to give a description of a symmetric monoidal triple category whose three types of morphisms, are lenses, charts, and what we call bisystems, that is morphisms - these are open dynamical systems which have two directions of interaction with the environment. We also describe a particular systems theory of smooth manifolds and smooth stochastic maps between them. Eigil Fjeldgren Rischel 2024 6 5 https://erischel.com/efr-0008/ efr-0008 /efr-0008/ Acknowledgements I would like to extend a great thanks to the Mathematically Structured Programming group at the University of Strathclyde, my fellow PhD students in particular, for making my time there such a great one. A good chunk of this work was completed while I worked at the Compositional Systems Laboratory at the Tallinn University of Technology---I owe everyone there a great thanks as well. For conversations which contributed to this work, and to my mathematical development in general, I would like to thank Dylan Braithwaite, Matteo Capucci, Elena di Lavore, Tobias Fritz, Davidad, Tomáš Gonda, Bruno Gavranović, Diana Kessler, Owen Lynch, Jade Master, David Jaz Myers, Riu Nakamura, Chad Nester, Evan Patterson, Paolo Perrone, Mario Roman, Brandon Shapiro, Toby Smithe, Pawel Sobocinski, David Spivak, and Andre Videla. A special thanks to Dylan, Matteo, and Bruno, for serving their time with me in the dependent optics mines. To Neil Ghani, Radu Mardare, and Jules Hedges, a massive thanks for serving as my advisors during the somewhat tumultuous process that was the production of this thesis. Finally, I thank my wonderful family. Now you can stop asking me about it. References Jules Hedges 2025 4 7 2015 3 20 https://erischel.com/hedges-string-games/ hedges-string-games /hedges-string-games/ String diagrams for game theory Reference 10.48550/arXiv.1503.06072 http://arxiv.org/abs/1503.06072 Jules Hedges 2025 4 7 2015 3 20 Abstract This paper presents a monoidal category whose morphisms are games (in the sense of game theory, not game semantics) and an associated diagrammatic language. The two basic operations of a monoidal category, namely categorical composition and tensor product, correspond roughly to sequential and simultaneous composition of games. This leads to a compositional theory in which we can reason about properties of games in terms of corresponding properties of the component parts. In particular, we give a definition of Nash equilibrium which is recursive on the causal structure of the game. The key technical idea in this paper is the use of continuation passing style for reasoning about the future consequences of players’ choices, closely based on applications of selection functions in game theory. Additionally, the clean categorical foundation gives many opportunities for generalisation, for example to learning agents. Matteo Capucci Owen Lynch David I. Spivak 2024 4 24 https://erischel.com/energy-driven-systems/ energy-driven-systems /energy-driven-systems/ Organizing Physics with Open Energy-Driven Systems Reference 10.48550/arXiv.2404.16140 http://arxiv.org/abs/2404.16140 Matteo Capucci Owen Lynch David I. Spivak 2024 4 24 Abstract Organizing physics has been a long-standing preoccupation of applied category theory, going back at least to Lawvere. We contribute to this research thread by noticing that Hamiltonian mechanics and gradient descent depend crucially on a consistent choice of transformation -- which we call a reaction structure -- from the cotangent bundle to the tangent bundle. We then construct a compositional theory of reaction structures. Reaction-based systems offer a different perspective on composition in physics than port-Hamiltonian systems or open classical mechanics, in that reaction-based composition does not create any new constraints that must be solved for algebraically. The technical contributions of this paper are the development of symmetric monoidal categories of open energy-driven systems and open differential equations, and a functor between them, functioning as a “functorial semantics” for reaction structures. This approach echoes what has previously been done for open games and open gradient-based learners, and in fact subsumes the latter. We then illustrate our theory by constructing an $n$-fold pendulum as a composite of $n$-many pendula. Jules Hedges Riu Rodriguez Sakamoto 2024 4 3 https://erischel.com/sakamoto-reinforcement-lenses/ sakamoto-reinforcement-lenses /sakamoto-reinforcement-lenses/ Reinforcement Learning in Categorical Cybernetics Reference 10.48550/arXiv.2404.02688 http://arxiv.org/abs/2404.02688 Jules Hedges Riu Rodriguez Sakamoto 2024 4 3 Abstract We show that several major algorithms of reinforcement learning (RL) fit into the framework of categorical cybernetics, that is to say, parametrised bidirectional processes. We build on our previous work in which we show that value iteration can be represented by precomposition with a certain optic. The outline of the main construction in this paper is: (1) We extend the Bellman operators to parametrised optics that apply to action-value functions and depend on a sample. (2) We apply a representable contravariant functor, obtaining a parametrised function that applies the Bellman iteration. (3) This parametrised function becomes the backward pass of another parametrised optic that represents the model, which interacts with an environment via an agent. Thus, parametrised optics appear in two different ways in our construction, with one becoming part of the other. As we show, many of the major classes of algorithms in RL can be seen as different extremal cases of this general setup: dynamic programming, Monte Carlo methods, temporal difference learning, and deep RL. We see this as strong evidence that this approach is a natural one and believe that it will be a fruitful way to think about RL in the future. Bruno Gavranović 2024 3 12 https://erischel.com/bruno-thesis-fundamental-components/ bruno-thesis-fundamental-components /bruno-thesis-fundamental-components/ Fundamental Components of Deep Learning: A category-theoretic approach Reference 10.48550/arXiv.2403.13001 http://arxiv.org/abs/2403.13001 Bruno Gavranović 2024 3 12 Abstract Deep learning, despite its remarkable achievements, is still a young field. Like the early stages of many scientific disciplines, it is marked by the discovery of new phenomena, ad-hoc design decisions, and the lack of a uniform and compositional mathematical foundation. From the intricacies of the implementation of backpropagation, through a growing zoo of neural network architectures, to the new and poorly understood phenomena such as double descent, scaling laws or in-context learning, there are few unifying principles in deep learning. This thesis develops a novel mathematical foundation for deep learning based on the language of category theory. We develop a new framework that is a end-to-end, b uniform, and c not merely descriptive, but prescriptive, meaning it is amenable to direct implementation in programming languages with sufficient features. We also systematise many existing approaches, placing many existing constructions and concepts from the literature under the same umbrella. In Part I we identify and model two main properties of deep learning systems parametricity and bidirectionality by we expand on the previously defined construction of actegories and Para to study the former, and define weighted optics to study the latter. Combining them yields parametric weighted optics, a categorical model of artificial neural networks, and more. Part II justifies the abstractions from Part I, applying them to model backpropagation, architectures, and supervised learning. We provide a lens-theoretic axiomatisation of differentiation, covering not just smooth spaces, but discrete settings of boolean circuits as well. We survey existing, and develop new categorical models of neural network architectures. We formalise the notion of optimisers and lastly, combine all the existing concepts together, providing a uniform and compositional framework for supervised learning. Pietro Vertechi 2023 8 7 https://erischel.com/vertechi-dep-optics/ vertechi-dep-optics /vertechi-dep-optics/ Dependent Optics Reference 10.4204/EPTCS.380.8 Pietro Vertechi 2023 8 7 Abstract A wide variety of bidirectional data accessors, ranging from mixed optics to functor lenses, can be formalized within a unique framework-dependent optics. Starting from two indexed categories, which encode what maps are allowed in the forward and backward directions, we define the category of dependent optics and establish under what assumptions it has coproducts. Different choices of indexed categories correspond to different families of optics: we discuss dependent lenses and prisms, as well as closed dependent optics. We introduce the notion of Tambara representation and use it to classify contravariant functors from the category of optics, thus generalizing the profunctor encoding of optics to the dependent case. Seth Frey Jules Hedges Joshua Tan Philipp Zahn 2023 3 9 https://erischel.com/hedges-etal-institutions/ hedges-etal-institutions /hedges-etal-institutions/ Composing games into complex institutions Reference 10.48550/arXiv.2108.05318 http://arxiv.org/abs/2108.05318 Seth Frey Jules Hedges Joshua Tan Philipp Zahn 2023 3 9 Abstract Game theory is used by all behavioral sciences, but its development has long centered around tools for relatively simple games and toy systems, such as the economic interpretation of equilibrium outcomes. Our contribution, compositional game theory, permits another approach of equally general appeal: the high-level design of large games for expressing complex architectures and representing real-world institutions faithfully. Compositional game theory, grounded in the mathematics underlying programming languages, and introduced here as a general computational framework, increases the parsimony of game representations with abstraction and modularity, accelerates search and design, and helps theorists across disciplines express real-world institutional complexity in well-defined ways. Relative to existing approaches in game theory, compositional game theory is especially promising for solving game systems with long-range dependencies, for comparing large numbers of structurally related games, and for nesting games into the larger logical or strategic flows typical of real world policy or institutional systems. Matteo Capucci Bruno Gavranović Jules Hedges Eigil Fjeldgren Rischel 2022 11 3 https://erischel.com/towards-cybercat/ towards-cybercat /towards-cybercat/ Towards Foundations of Categorical Cybernetics Reference 10.4204/EPTCS.372.17 Matteo Capucci Bruno Gavranović Jules Hedges Eigil Fjeldgren Rischel 2022 11 3 Abstract We propose a categorical framework for processes which interact bidirectionally with both an environment and a “controller”. Examples include open learners, in which the controller is an optimiser such as gradient descent, and an approach to compositional game theory closely related to open games, in which the controller is a composite of game-theoretic agents. We believe that “cybernetic” is an appropriate name for the processes that can be described in this framework. Dylan Braithwaite Jules Hedges 2022 9 29 https://erischel.com/hedges-braithwaite-dep/ hedges-braithwaite-dep /hedges-braithwaite-dep/ Dependent Bayesian Lenses: Categories of Bidirectional Markov Kernels with Canonical Bayesian Inversion Reference 10.48550/arXiv.2209.14728 http://arxiv.org/abs/2209.14728 Dylan Braithwaite Jules Hedges 2022 9 29 Abstract We generalise an existing construction of Bayesian Lenses to admit lenses between pairs of objects where the backwards object is dependent on states on the forwards object (interpreted as probability distributions). This gives a natural setting for studying stochastic maps with Bayesian inverses restricted to the points supported by a given prior. In order to state this formally we develop a proposed definition by Fritz of a support object in a Markov category and show that these give rise to a section into the category of dependent Bayesian lenses encoding a more canonical notion of Bayesian inversion. Andre Videla Matteo Capucci 2022 3 29 https://erischel.com/videla-capucci-servers/ videla-capucci-servers /videla-capucci-servers/ Lenses for Composable Servers Reference 10.48550/arXiv.2203.15633 http://arxiv.org/abs/2203.15633 Andre Videla Matteo Capucci 2022 3 29 Abstract We implement the semantics of server operations using parameterised lenses. They allow us to define endpoints and extend them using classical lens composition. The parameterised nature of lenses models state updates while the lens laws mimic properties expected from HTTP. This first approach to server development is extended to use dependent parameterised lenses. An upgrade necessary to model not only endpoints, but entire servers, unlocking the ability to compose them together. Dylan Braithwaite Matteo Capucci Bruno Gavranović Jules Hedges Eigil Fjeldgren Rischel 2021 12 21 https://erischel.com/fibre-optics-2021/ fibre-optics-2021 /fibre-optics-2021/ Fibre optics Reference 10.48550/arXiv.2112.11145 http://arxiv.org/abs/2112.11145 Dylan Braithwaite Matteo Capucci Bruno Gavranović Jules Hedges Eigil Fjeldgren Rischel 2021 12 21 Abstract Lenses, optics and dependent lenses (or equivalently morphisms of containers, or equivalently natural transformations of polynomial functors) are all widely used in applied category theory as models of bidirectional processes. From the definition of lenses over a finite product category, optics weaken the required structure to actions of monoidal categories, and dependent lenses make use of the additional property of finite completeness (or, in case of polynomials, even local cartesian closure). This has caused a split in the applied category theory literature between those using optics and those using dependent lenses. The goal of this paper is to unify optics with dependent lenses, by finding a definition of fibre optics admitting both as special cases. 2021 12 7 https://erischel.com/milewski-polylens/ milewski-polylens /milewski-polylens/ PolyLens Reference https://bartoszmilewski.com/2021/12/07/polylens/ G. S. H. Cruttwell Bruno Gavranović Neil Ghani Paul Wilson Fabio Zanasi 2021 7 13 https://erischel.com/bruno-etal-categorical-learning-2021/ bruno-etal-categorical-learning-2021 /bruno-etal-categorical-learning-2021/ Categorical Foundations of Gradient-Based Learning Reference 10.48550/arXiv.2103.01931 http://arxiv.org/abs/2103.01931 G. S. H. Cruttwell Bruno Gavranović Neil Ghani Paul Wilson Fabio Zanasi 2021 7 13 Abstract We propose a categorical semantics of gradient-based machine learning algorithms in terms of lenses, parametrised maps, and reverse derivative categories. This foundation provides a powerful explanatory and unifying framework: it encompasses a variety of gradient descent algorithms such as ADAM, AdaGrad, and Nesterov momentum, as well as a variety of loss functions such as as MSE and Softmax cross-entropy, shedding new light on their similarities and differences. Our approach to gradient-based learning has examples generalising beyond the familiar continuous domains (modelled in categories of smooth maps) and can be realized in the discrete setting of boolean circuits. Finally, we demonstrate the practical significance of our framework with an implementation in Python. Elena Di Lavore Jules Hedges Pawel Sobocinski 2021 https://erischel.com/hedges-etal-graph-games/ hedges-etal-graph-games /hedges-etal-graph-games/ Compositional modelling of network games Reference 10.4230/LIPIcs.CSL.2021.30 Elena Di Lavore Jules Hedges Pawel Sobocinski 2021 Abstract The analysis of games played on graph-like structures is of increasing importance due to the prevalence of social networks, both virtual and physical, in our daily life. As well as being relevant in computer science, mathematical analysis and computer simulations of such distributed games are vital methodologies in economics, politics and epidemiology, amongst other fields. Our contribution is to give compositional semantics of a family of such games as a well-behaved mapping, a strict monoidal functor, from a category of open graphs (syntax) to a category of open games (semantics). As well as introducing the theoretical framework, we identify some applications of compositionality. Joe Bolt Jules Hedges Philipp Zahn 2019 10 8 https://erischel.com/hedges-etal-bayesian-games/ hedges-etal-bayesian-games /hedges-etal-bayesian-games/ Bayesian open games Reference http://arxiv.org/abs/1910.03656 Joe Bolt Jules Hedges Philipp Zahn 2019 10 8 Abstract This paper generalises the treatment of compositional game theory as introduced by the second and third authors with Ghani and Winschel, where games are modelled as morphisms of a symmetric monoidal category. From an economic modelling perspective, the existing notion of an open game is not expressive enough for many applications. This includes stochastic environments, stochastic choices by players, as well as incomplete information regarding the game being played. The current paper addresses these three issue all at once. To achieve this we make significant use of category theory, especially the "coend optics" of Riley. Brendan Fong David I. Spivak Rémy Tuyéras 2019 5 1 https://erischel.com/backprop-as-functor/ backprop-as-functor /backprop-as-functor/ Backprop as Functor: A compositional perspective on supervised learning Reference http://arxiv.org/abs/1711.10455 Brendan Fong David I. Spivak Rémy Tuyéras 2019 5 1 Abstract A supervised learning algorithm searches over a set of functions A → B parametrised by a space P to find the best approximation to some ideal function f : A → B. It does this by taking examples (a, f (a)) ∈ A × B, and updating the parameter according to some rule. We define a category where these update rules may be composed, and show that gradient descent—with respect to a fixed step size and an error function satisfying a certain property—defines a monoidal functor from a category of parametrised functions to this category of update rules. A key contribution is the notion of request function. This provides a structural perspective on backpropagation, giving a broad generalisation of neural networks and linking it with structures from bidirectional programming and open games. Jules Hedges 2018 8 16 https://erischel.com/jules-lenses-blogpost/ jules-lenses-blogpost /jules-lenses-blogpost/ Lenses for philosophers Reference https://julesh.com/2018/08/16/lenses-for-philosophers/ Neil Ghani Jules Hedges Viktor Winschel Philipp Zahn 2018 2 15 https://erischel.com/hedges-etal-comp-gametheory/ hedges-etal-comp-gametheory /hedges-etal-comp-gametheory/ Compositional game theory Reference 10.48550/arXiv.1603.04641 http://arxiv.org/abs/1603.04641 Neil Ghani Jules Hedges Viktor Winschel Philipp Zahn 2018 2 15 Abstract We introduce open games as a compositional foundation of economic game theory. A compositional approach potentially allows methods of game theory and theoretical computer science to be applied to large-scale economic models for which standard economic tools are not practical. An open game represents a game played relative to an arbitrary environment and to this end we introduce the concept of coutility, which is the utility generated by an open game and returned to its environment. Open games are the morphisms of a symmetric monoidal category and can therefore be composed by categorical composition into sequential move games and by monoidal products into simultaneous move games. Open games can be represented by string diagrams which provide an intuitive but formal visualisation of the information flows. We show that a variety of games can be faithfully represented as open games in the sense of having the same Nash equilibria and off-equilibrium best responses. Mitchell Riley 2018 https://erischel.com/riley-optics/ riley-optics /riley-optics/ Categories of optics Reference Jm Hedges 2016 10 3 https://erischel.com/hedges-towards-compositional-thesis/ hedges-towards-compositional-thesis /hedges-towards-compositional-thesis/ Towards compositional game theory Reference https://qmro.qmul.ac.uk/xmlui/handle/123456789/23259 Jm Hedges 2016 10 3 Abstract We introduce a new foundation for game theory based on so-called open games. Unlike existing approaches open games are fully compositional: games are built using algebraic operations from standard components, such as players and outcome functions, with no fundamental distinction being made between the parts and the whole. Open games are intended to be applied at large scales where classical game theory becomes impractical to use, and this thesis therefore covers part of the theoretical foundation of a powerful new tool for economics and other subjects using game theory. Formally we defi ne a symmetric monoidal category whose morphisms are open games, which can therefore be combined either sequentially using categorical composition, or simultaneously using the monoidal product. Using this structure we can also graphically represent open games using string diagrams. We prove that the new de definitions give the same results (both equilibria and o -equilibrium best responses) as classical game theory in several important special cases: normal form games with pure and mixed strategy Nash equilibria, and perfect information games with subgame perfect equilibria. This thesis also includes work on higher order game theory, a related but simpler approach to game theory that uses higher order functions to model players. This has been extensively developed by Martin Escard o and Paulo Oliva for games of perfect information, and we extend it to normal form games. We show that this approach can be used to elegantly model coordination and di differentiation goals of players. We also argue that a modify cation of the solution concept used by Escard o and Oliva is more appropriate for such applications. David Jaz Myers https://erischel.com/myers-cst/ myers-cst /myers-cst/ Categorical Systems Theory Reference http://davidjaz.com/Papers/DynamicalBook.pdf Owen Lynch Eigil Fjeldgren Rischel David Jaz Myers Sam Staton https://erischel.com/lynch-myers-rischel-staton-stoch-clocks/ lynch-myers-rischel-staton-stoch-clocks /lynch-myers-rischel-staton-stoch-clocks/ Clock systems for stochastic and non-deterministic categorical systems theories Reference Context Eigil Fjeldgren Rischel 2024 5 25 https://erischel.com/efr-0001/ efr-0001 /efr-0001/ Markov Fibrations Eigil Fjeldgren Rischel 2025 4 6 https://erischel.com/efr-HQ73/ efr-HQ73 /efr-HQ73/ Introduction In applying category theory to fields as diverse as game theory (Reference ,Reference , Reference ), machine learning (Reference , Reference ), dynamical systems (Reference , Reference ), and server design (Reference ), people have found it useful to study categories whose morphisms describe processes or functions that take place in two "stages", where the second is dependent on the first, but composes in the other direction---so-called lenses. In many of these domains, stochastic phenomena play a role. There is a useful generalization of lenses to categories of stochastic maps---the so-called optics of Riley, Reference ---but this does not accommodate another useful generalization, so called dependent lenses where not only the backwards process but the set it takes values in is indexed over the base. It has been a long-standing problem to develop a suitable common generalization, "dependent optics", of these two ideas. In this thesis, we solve this problem by developing a theory of Markov fibrations (§ ). A Markov fibration is a weakening of the notion of (Grothendieck) fibration to include (subject to some assumptions) Markov categories of indexed families of objects (given by deterministic functions ) and compatible stochastic maps (given by commutative squares). The main point of Markov fibrations is that they admit fiberwise opposites, which generalize the fiberwise opposites of ordinary categories. Just as the fiberwise opposite of the codomain fibration of a finitely complete category describes dependent lenses (see § ), the fiberwise opposite of these codomain Markov fibrations give a good notion of stochastic lens (see Theorem ). We will apply our theory chiefly to two problems. First, we will use them to generalize open games (Hedges, Reference ) to a larger class of interfaces (namely, indexed families of sets) while at the same time considering possibly-stochastic maps. This will allow us to construct the so-called external choice operator on open games, which describes branching. Secondly, we will generalize Myers' categorical dynamical systems theory to allow for stochastic maps in the base. When combined with another generalization of these systems, to general parametrized lenses, this gives a more natural way of modeling certain stochastic dynamical systems, such as those associated to the training dynamics of machine learning models, see eg Example . Eigil Fjeldgren Rischel 2025 4 6 A brief introduction to lenses If is a category with finite products, the category of lenses in has objects pairs of objects in , and morphisms given by pairs . (Note that it is of course the morphisms, not the objects, that are called lenses). One way to generalize this is to ask for to have pullbacks, and consider an object given by a more general map , and let a map be (so that the triangle over commutes). (Note that this recovers the "simple lenses" when the objects are of the form ). Under the interpretation of a map as a family of objects indexed over the elements of , we see this as a "dependent lens" (since is a "dependent type"). There are various variations of this idea, which ultimately all fit the pattern of taking a Grothendieck fibration and forming the fiberwise opposite---for the dependent lenses as above, this is the codomain fibration . Meanwhile, another wide-ranging generalization of are the optics of Riley (Reference ). For a monoidal category, the objects of are again pairs of objects, but now the morphisms are elements of the coend That is, to give an optic is to give an object and morphisms , up to the equivalence relation generated by, whenever identifying the two optics given by and . One can show that, in the case where is a Cartesian monoidal category, this set can be identified with the set of (simple) lenses. A vector field on a smooth manifold is simply a (smooth) section of the tangent bundle . A section is a bundle map from the trivial bundle . This gives the idea of considering a parametrized dynamical system as consisting of a map to some other smooth manifold, a bundle (the points of are the parameters) and a bundle map . As we will see, such a bundle map is exactly a dependent lens in a category of bundles. Building on this idea, Myers (Reference ) described a highly abstract theory of open dynamical systems, given by lenses out of tangent bundles (where the notion of space, bundle and tangent bundle are generalized to any fibration). It has been observed that, although a lens from the tangent bundle may indeed be said to describe an open dynamical system with state space , the same can be said for a lens of type ---although a dynamical system of a different kind (essentially, the first type are the Moore machines, the second the Mealy machines---see the introduction to § for more on this). It is natural to consider parametrized maps as a common generalization of these two concepts---and in fact, special cases of this idea, lenses parametrized by tangent bundles, have already been considered many times, for example in the semantics of gradient descent (see Reference , Reference , Reference ). Since parametrized maps compose in an obvious way, we now have three different notions of morphisms between bundles---lenses, charts, and "generalized systems". These should form some sort of symmetric monoidal triple category, but the right axiomatisation of this concept is somewhat elusive. Lenses, in their various incarnations, have been widely used in applied category theory. For example, Hedges and his collaborators have developed a compositional approach to game theory (Reference , Reference , Reference , Reference , others). Recent work by Hedges and Sakamoto bring this idea to reinforcement learning (Reference ). We briefly mentioned Myers' categorical systems theory above, and we will see much more of it later. As we discussed above, there is also a literature using lenses to study gradient descent in an abstract sense. This is without even mentioning their original role in the theory of functional programming (this is the origin of the somewhat confusing term lens), or their prehistory in Gödel's dialectica interpretation---see eg Reference for a survey of this. Eigil Fjeldgren Rischel 2025 4 6 Dependent Optics An open game, in the sense of Hedges, has as its interface two objects in the category of lenses (of sets) (that is, two pairs of sets). An open game with this interface consists of a set of strategies equipped with a function and a subset of equilibrium strategies for each and ---note that such an is exactly a lens and such a is exactly a lens . This view of open games makes the composition rule much simpler to define, although we will not delve into the details here, see eg Reference . The idea is that the open game represents the strategies and preferences of a player or set of players--- are the possible strategies, is the information revealed to the player before they make their decision of which moves to make, the resulting is the choice the player "sends" in response to , and the is the "utility", the value they eventually learn, depending on their choice , which they have some preferences over. The above description gives a theory of deterministic games, but of course, it is completely essential for game theory to model both random decisions (mixed strategies) and decisions made under uncertainty. This motivates the replacement of lenses in this definition with optics in a category of probability kernels (the form of the equilibrium relation must be modified as well, see Reference , Reference ) In the abstract study of open games, it is desirable to define a "choice" operation on the objects (the pairs of sets) so that maps into represent players who are faced with some binary choice between and , after which play may proceed for a time in one of two branches. However, this is immediately problematic because the category of lenses do not have coproducts---inside the category of dependent lenses, however, we can form the coproduct simply as ---that is, we get a family which is over , and over . This indeed does the job, but for game theory the extension to stochastic maps---optics---is absolutely essential. This motivates the search for a theory of dependent optics, a common generalization of optics and dependent lenses. There have been a number of attempts at this---see eg Reference , Reference , Reference , Reference . While these efforts have to some extent succeeded in describing a theory that is general enough to contain the desired examples, it is generally very ad hoc. The construction of optics as a special case of Vertechi's dependent optics (Reference ) gives them as fibre optics with the indexing bicategory being (the delooping of) a monoidal category , but embeds lenses using a bicategory of spans. Thus simple lenses admit two distinct encodings in this theory. Moreover, Vertechi's example of dependent monoidal optics fails to describe a supercategory of when applied to Markov categories, and are thus not suitable for, for example, game theory. Eigil Fjeldgren Rischel 2025 4 6 Categorical Cybernetics The observation that both open games and machine learning systems can be fruitfully described using lenses, naturally led to the idea that this could be used to develop a more thorough analogy between the two. After all, a machine learning system is also a player in a kind of game (with the objective of minimizing loss). Hedges coined the phrase categorical cybernetics to describe this locus of ideas (Reference , also used in Reference ). Myers' categorical systems theory also describes dynamical systems as lenses. However, where a cybernetic system in the sense of Reference is a parametrized lens and thus has two different inputs - which is the information on which basis they are allowed to choose their decision and which is their "utility", the information they are allowed to care about. It is straightforward to describe the machine learning part of the categorical cybernetics analogy in terms of Myers' theory---a "learner" is simply a lens of the above form with replace with a tangent bundle . This idea (described in these terms, although not with the analogy to machine learning) has already been described by Reference . However, for game theory, it is necessary that the output of a given system can be stochastically chosen (a player must be able to randomize their strategy). At the same time, for a machine learning system, the is usually some sort of sample from a training distribution---ie, it is random. Hence to really describe the training dynamics of machine learning systems using this theory, we are again naturally drawn to consider systems theories (that is, fibrations) which have stochastic maps in the base, not merely the fiber. Eigil Fjeldgren Rischel 2025 4 6 Overview and contributions We begin the thesis in § with a review of some preliminary material. This chapter can largely be skipped for readers already familiar with the material (Markov categories, double categories, lenses and optics, fibrations, and categorical dynamical systems theory). The chapter on Markov categories introduces a few novel auxiliary notions, but these can be referred back to as necessary (they are primarily regularity conditions which hold in most Markov categories of interest). In § , we give the main technical contribution of the thesis by developing a theory of Markov fibrations. Their fiberwise opposites generalize both the fiberwise opposites of codomain fibrations (dependent lenses in a classical sense) and optics in Markov categories. Theorem provides the statement of the latter (the former is straightforward). We also give a description of monoidal structures on Markov fibrations (which pass to monoidal structures on the resulting categories of optics). In § , we give a construction of the double category of parametrized morphisms for a category action (or actegory)---in fact, we do this internally to any -category. This has the advantage of allowing us to construct more highly-structured versions of this double category by carrying out the construction internally to higher-structured actions. (Bicategorical versions of are a relatively old idea, and the double categorical version is a straightforward extension which has existed for some time as folklore, but the fully-internal construction here is novel). In § , we apply Markov fibrations to compositional game theory, and solve a longstanding problem by giving a category of stochastic open games with a general "external choice" operator. This construction is largely abstract over the particular category, and so can be applied to different Markov fibrations to describe games defined with different notions of stochasticity. We also review some previous work with Capucci, Hedges, and Gavranovic on abstract constructions of categories of open games (Reference ). In § , we describe how to extend Myers' categorical systems theory to Markov fibrations, to give theories of systems which may have stochastic maps not merely in the fiber (as was already described by Myers in Reference ) but also in the base. We also leverage our construction from § to give a description of a symmetric monoidal triple category whose three types of morphisms, are lenses, charts, and what we call bisystems, that is morphisms - these are open dynamical systems which have two directions of interaction with the environment. We also describe a particular systems theory of smooth manifolds and smooth stochastic maps between them. Eigil Fjeldgren Rischel 2024 6 5 https://erischel.com/efr-0008/ efr-0008 /efr-0008/ Acknowledgements I would like to extend a great thanks to the Mathematically Structured Programming group at the University of Strathclyde, my fellow PhD students in particular, for making my time there such a great one. A good chunk of this work was completed while I worked at the Compositional Systems Laboratory at the Tallinn University of Technology---I owe everyone there a great thanks as well. For conversations which contributed to this work, and to my mathematical development in general, I would like to thank Dylan Braithwaite, Matteo Capucci, Elena di Lavore, Tobias Fritz, Davidad, Tomáš Gonda, Bruno Gavranović, Diana Kessler, Owen Lynch, Jade Master, David Jaz Myers, Riu Nakamura, Chad Nester, Evan Patterson, Paolo Perrone, Mario Roman, Brandon Shapiro, Toby Smithe, Pawel Sobocinski, David Spivak, and Andre Videla. A special thanks to Dylan, Matteo, and Bruno, for serving their time with me in the dependent optics mines. To Neil Ghani, Radu Mardare, and Jules Hedges, a massive thanks for serving as my advisors during the somewhat tumultuous process that was the production of this thesis. Finally, I thank my wonderful family. Now you can stop asking me about it. Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-POD2/ efr-POD2 /efr-POD2/ Preliminaries Eigil Fjeldgren Rischel 2025 5 6 Introduction We will begin by reviewing some theory that plays a key role in this thesis. With a very few exceptions, nothing here is novel, but we find it useful to include these here---both for the convenience of the reader, to familiarize them with theory that we will make constant reference to, but also to set the stage for our contributions. First, the theory of (Grothendieck) fibrations. There is far too much to say about these for such a brief space, so we will limit ourselves to what we need for the rest of the thesis, especially for the section on Markov fibrations. Second, we will give an overview of optics and lenses, in a bit more detail than the introduction. We have already given a review of the sources there, but we will find it useful to put this on proper footing. Next, we will review the theory of Markov categories. This is a synthetic approach to probability theory, introduced by Fritz Reference , and since developed further by many collaborators, including the author. Here we note the only exceptions to the claim that nothing in this chapter is novel. First, the notion of representable Markov category Definition was introduced by Fritz, Gonda, Perrone, and the author in Reference . We will not give a thorough treatment here, but since representable Markov categories are so ubiquitous, we will frequently note how different properties or structure on a Markov category relates to representability. We will also introduce a few new concepts which play a role in the theory of Markov fibrations in § . These are of no great independent interest, as far as we can tell, nor are they difficult, but this seemed the best place to put them. Finally, we give a brief review of Myers' categorical systems theory. This will mainly be to set the stage for § , where we develop a triple categorical version of the theory. Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-3AEA/ efr-3AEA /efr-3AEA/ Fibrations In category theory, there are many families of categories indexed by the objects of some other category. For example, for each commutative ring, we have the category . Given a ring homomorphism there is an induced restriction of scalars functor (given simply by composing the module structure by ), and this is (contravariant) functorial, assembling into a functor . In most cases, one can not expect strict functoriality as above. From an abstract point of view, it makes sense that one should really ask only for a natural isomorphism , up to some coherence conditions. This assembles into a so-called pseudofunctor into the 2-category . From a concrete point of view, there are many natural families of categories which arise as pseudofunctors. For example, restriction of scalars always has a left adjoint (extension of scalars, given by viewing as an -module via the map )---since adjoints compose (that is, if and then ) this must be functorial up to natural isomorphism, but this is the best we can promise. To avoid the higher categorical algebra involved in working with pseudofunctors, Grothendieck introduced the notion of fibration in Reference . Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-Z7EF/ efr-Z7EF /efr-Z7EF/ Definition Let be a functor. Given , write for the (strict) pullback . Explicitly, this consists of the objects in with and the morphisms with . Let be a morphism. A map with is locally Cartesian if for each with , postcomposition with induces a bijection A map is Cartesian if for every and with there is a bijection note that every Cartesian map is locally Cartesian (take ) is a Grothendieck fibration (or just fibration) if, for every such that , there exists a Cartesian map (for some ) so that Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-Z4L4/ efr-Z4L4 /efr-Z4L4/ Lemma is a Grothendieck fibration if and only if every admits a locally Cartesian lift, and the class of locally Cartesian morphisms in is stable under composition. Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-QHXA/ efr-QHXA /efr-QHXA/ Theorem Let be a Grothendieck fibration. For every select a Cartesian lift of . Then there is a unique extension of to a functor so that the squares commute. With this, the assignment assembles into a pseudofunctor . Eigil Fjeldgren Rischel 2025 5 5 https://erischel.com/efr-WKG8/ efr-WKG8 /efr-WKG8/ Remark If is such that each morphism admits merely a locally Cartesian lift (but these do not compose,) it is called a prefibration. Note that this is unrelated to our notion of Markov prefibration (reading ahead a bit, the "Cartesian" maps in a Markov prefibration do compose, but they enjoy the unique lifting property only for a subset of morphisms). This clash of terminology is perhaps unfortunate, but other potential prefixes seemed inferior (quasi-, pseudo-, semi-). Eigil Fjeldgren Rischel 2025 5 7 https://erischel.com/efr-D7W0/ efr-D7W0 /efr-D7W0/ Example Let be any category, and let denote the arrow category. Then the codomain functor is a fibration if and only if admits all pullbacks, and in this case the functors given are given by pullback along . The functors are sometimes referred to as pullback, a convention we generally adopt. They are also sometimes called base-change functors. When and , we may write for the object if there is no chance of confusion. (Compare that the choice of is also suppressed in the notation for a pullback) We will not go into a comprehensive description of the theory of fibrations, but simply give a few basic results. We will give some examples in the next section. For a textbook treatment, see eg. Reference (chapters 1, 9), or Reference (chapter 8). Note that we will not give a formal definition of the term "pseudofunctor" here. See eg Reference , def. 1.4.4. Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-KUW7/ efr-KUW7 /efr-KUW7/ Definition Let be a pseudofunctor. Then there is a category defined as follows: The objects are pairs The morphisms are pairs Composition is given by the "chain rule" There is an obvious forgetful functor . The category is known as the Grothendieck construction of Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-TQ3W/ efr-TQ3W /efr-TQ3W/ Proposition is a Grothendieck fibration, and the pseudofunctor it induces is equivalent to Eigil Fjeldgren Rischel2025428Proof Given and it is clear that the map given by is locally Cartesian---the required bijection is the definition of maps in the Grothendieck construction. But it's straightforward to see that these compose. Given a pseudofunctor it is obvious that the assignment is pseudofunctorial as well (the required natural isomorphisms are just the formal opposites of the ones for ). Applying this through the equivalence of fibrations and pseudofunctors leads to the fiberwise opposite of a fibration. Explicitly: Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-8LGE/ efr-8LGE /efr-8LGE/ Corollary Let be a fibration. Then there exists a category , called the fiberwise opposite of , whose objects are the same as , and where a morphism is a tuple . Eigil Fjeldgren Rischel 2025 4 29 https://erischel.com/efr-ZCTD/ efr-ZCTD /efr-ZCTD/ Optics and Lenses We have already given somewhat of an account of lenses, optics, and their applications in the introduction. We briefly review the theory here, especially to normalize the notation and definitions. The best general source for this material is still Reference . The definition of optic relies on the notion of coend, which we briefly recall, see Reference for a textbook account. If is a functor, the coend is defined as the initial object receiving a map for each , so that for each , the square commutes. If has enough colimits, we may express the coend as the coequalizer of the diagram where the two maps are given on each component by and , respectively. (This is Remark 1.2.4 in Reference ). Note that in particular this implies that if has all colimits, then it also has all coends. Eigil Fjeldgren Rischel 2025 4 29 https://erischel.com/efr-MMQZ/ efr-MMQZ /efr-MMQZ/ Optic Definition Let be a monoidal category which acts on two categories . Then the category of optics has Objects pairs The set of morphisms given by the coend Given two optics with representatives their composite is given by where we omit coherence morphisms. When is a monoidal category acting on itself by tensor, we write Note that if are symmetric monoidal and these actions are symmetric, inherits a symmetric monoidal structure given by . (Also given a braiding one can induce a non-symmetric monoidal structure, but this almost never comes up). Eigil Fjeldgren Rischel 2025 4 29 https://erischel.com/efr-CRYH/ efr-CRYH /efr-CRYH/ On notation Remark Objects and morphisms in the category of optics have two parts---one going "forwards", in the same direction as the optic, and one going "backwards". In Reference the objects are written where is the forwards part. Hedges' work on open games used the binomial notation but wrote the forwards part on top. To make the connection to fibrations more natural, we instead write the forwards part on the bottom, . It is the backwards part which depends on the forwards part, hence the forwards part is the base of the fibration (when one exists)---and every part of the language of fibrations is built around a mental model where the base is at the bottom and the fibers are over it (including the word "base"). When reading the references, this may cause some confusion, but hopefully this can be overcome. While we're at it, let us note that when talking about optics we will freely use terms like "the forwards part" "the backwards object" and so on---the meaning of this should now be clear. Of course, once we get to cooptics/charts, this would be more than a little confusing, since in that case both components are in the same direction. In those cases we will speak of either the primary (forwards) part or the secondary (backwards) part, or use the language of fibrations and speak of the map or object "in the base" and "in the fiber". As noted above, the coend consists of triples up to the equivalence relation which, for every identifies the two tuples and . Note that this relation is not assumed to be inherently an equivalence relation---one takes the transitive-symmetric closure as usual. We call this relation the sliding relation (because we slide the map from the backwards part to the forwards part). Eigil Fjeldgren Rischel 2025 8 25 https://erischel.com/efr-50T5/ efr-50T5 /efr-50T5/ Remark Suppose are small categories. Then the coend defining is a small colimit of small sets, hence again small. Since clearly the set of objects is small, is again a small category. However, if are merely assumed to be locally small, we can not guarantee the same is true of , since the hom-sets are now defined by a coend/colimit with large indexing category. However, in many special cases, it can still be seen to be locally small, such as in the Cartesian case (where is clearly locally small). In this thesis, we will not delve further into this subtlety, simply working inside some universe where all our categories are small. Of course, we can also let the arrows in go in the same direction as . This does not seem to have played any role in the literature, but we will give this a name, as it is a useful example to have in mind for Markov fibrations (where we will construct our dependent optics, conceptually, as a fiberwise opposite) Eigil Fjeldgren Rischel 2025 4 29 https://erischel.com/efr-G0YN/ efr-G0YN /efr-G0YN/ Co-Optics Definition Given acting on , the category of co-optics, has objects pair and morphisms given by the coend Eigil Fjeldgren Rischel 2025 5 7 https://erischel.com/efr-J0WW/ efr-J0WW /efr-J0WW/ Remark For any monoidal category , . One way to think of an optic is as a string diagram , but which has a hole with space for a morphism . One inserts such a morphism by composing the optic with the optic representing it. This idea of "open diagrams" has been developed in much more detail by Román, Reference . Eigil Fjeldgren Rischel 2025 5 7 https://erischel.com/efr-UNIN/ efr-UNIN /efr-UNIN/ Proposition Suppose is semicartesian---in other words, that is terminal. Then there is a functor , which takes a to , and pair to the composite . In the case of the map is a bijection. The fact that in states (maps from the monoidal unit) on are given by states on , while costates are given by maps plays an important role in the use of optics to describe open games. See § for more on this. Let us say a few things about lenses. Eigil Fjeldgren Rischel 2025 5 7 https://erischel.com/efr-M19V/ efr-M19V /efr-M19V/ Proposition If is Cartesian monoidal, Eigil Fjeldgren Rischel202557Proof Here we use a general fact about coends, that ---this has been called the ninja Yoneda lemma, see Reference . In this case we sometimes write for the category . We have the following fact: Eigil Fjeldgren Rischel 2025 5 7 https://erischel.com/efr-2793/ efr-2793 /efr-2793/ Proposition The functor , , is a fibration. The fiber has the following description: Its objects are the objects of . A map is a map . The composite of is given by composing the two into then using the diagonal. Given the pullback functor is given by precomposing by . Eigil Fjeldgren Rischel 2025 5 7 https://erischel.com/efr-VTPS/ efr-VTPS /efr-VTPS/ Proposition If moreover admits pullbacks, there is a fibred functor which carries an object to and a morphism to the map over . This is fully faithful. This is the first way to see the maps of as dependent lenses---they receive the category of lenses as a full subcategory. Eigil Fjeldgren Rischel 2025 5 7 https://erischel.com/efr-TZ0P/ efr-TZ0P /efr-TZ0P/ Remark The squares are pullbacks in any category with products, even if it does not admit pullbacks in general. It follows that the full subcategory of spanned by objects of this form is always a fibration over , which is isomorphic to ---the fiberwise dual is isomorphic to . Eigil Fjeldgren Rischel 2025 4 18 https://erischel.com/efr-E3HE/ efr-E3HE /efr-E3HE/ Markov categories Recall that Markov categories (Fritz, Reference ) are semicartesian symmetric monoidal categories equipped with a suitably coherent system of comonoids for each object . These have been used as the basis for the categorical analysis of probability theory for a number of years, see eg Reference , Reference , Reference , Reference . See Reference for the basic theory, and Reference for the theory of representable Markov categories. We assume familiarity with this theory in general, but we will review some theory below, as well as a few new minor results which are helpful for the theory of Markov fibrations. The basic idea of a Markov category is to interpret morphisms as "stochastic processes" or kernels---that is, functions valued in probability measures. A morphism is a parametrized joint probability measure---the comonoid structure allows us to build a canonical such given parametrized measures (by precomposing their tensor with the map). This is the product measure of the two---the fact that in general not every map has this form (because is not necessarily Cartesian) allows us to express the probabilistic dependence---as in, non-independence---of one variable on another. A morphism is called deterministic if it is a comonoid (co)homomorphism, which amounts to the claim that ---in other words, that running two independent copies of the kernel (with the same input) is equivalent to running one and copying the output. The deterministic morphisms form a Cartesian monoidal subcategory which is denoted . We first note the following alternative characterization of Markov categories in terms of their deterministic morphisms. Eigil Fjeldgren Rischel 2025 5 2 https://erischel.com/efr-5B0J/ efr-5B0J /efr-5B0J/ Premarkov structure Definition Let be a symmetric monoidal category. A premarkov structure on is a wide symmetric monoidal subcategory ---that is, a class of morphisms which contains all identities and structural isomorphisms, and is stable under composition and monoidal products---so that the monoidal category is Cartesian. Eigil Fjeldgren Rischel 2025 5 2 https://erischel.com/efr-10HQ/ efr-10HQ /efr-10HQ/ Proposition Let be a monoidal category. Given a premarkov structure , there is a unique Markov structure on so that each morphism in is deterministic. Given a Markov structure, is a premarkov structure. A premarkov structure has the form for some Markov structure if and only if it is maximal. Eigil Fjeldgren Rischel202552Proof The existence part first claim is clear: acquires a unique Markov structure since it is Cartesian, and the inclusion of that Markov structure into gives a Markov structure on . Conversely, suppose is a Markov category and is a class of deterministic morphisms which is still Cartesian. This means the projections still exhibit as a product in . But since the pairing are the unique map lifting two given maps the pairing must be preserved by the inclusion . Since the canonical Markov structure is given as a pairing, this means the markov structure induced by must agree with the one given by , which is just the original one. The second claim is straightforward---see eg Reference remark 10.13. Now suppose , where is some larger premarkov structure. By the argument above, they must generate the same Markov structure on . But again, this implies that every map in is deterministic for this Markov structure, so we have . This finishes the proof. Note that given a merely monoidal category, we can ask whether it is Cartesian, and if it is, it admits a unique symmetry induced by the universal property of the product. Hence if is a Cartesian wide monoidal subcategory, we may attempt to define a symmetry on simply using the one from . However, it is not automatic that this symmetry is natural for all the morphisms in . This basic idea was already noted by Fritz (and goes back to Golubtsov's work in Reference , an important part of the prehistory of Markov categories), although the precise statement above appears to be novel. Since the coherence conditions required of the comonoids in a Markov structure can be somewhat hard to remember, this characterization may be easier to understand. Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-38FR/ efr-38FR /efr-38FR/ Representable Markov category Definition A Markov category is representable if the inclusion admits a right adjoint. In this case we denote the right adjoint and call the object for a distribution object for . Observe that (by definition,) and hence where we denote the induced monad on by an abuse of notation. Eigil Fjeldgren Rischel 2025 4 27 https://erischel.com/efr-OSN1/ efr-OSN1 /efr-OSN1/ Commonly used Markov Categories Example Here are some important Markov categories: (Reference , section 4) is the category whose objects are measurable spaces, whose morphisms are Markov kernels, with composition given by the Chapman-Kolmogorov equation and monoidal structure given by product measures. (Reference , section 4) is the full subcategory of spanned by the standard Borel spaces, that is by those measurable spaces arising as the Borel -algebra on separable, complete metric space. is the subcategory of spanned by finite sets in the powerset -algebra. A morphism in is equivalently a matrix with entries in and with for each (this is what is called a stochastic matrix). Let be the monad which assigns to the set of countably-supported probability measures. Then the Kleisli category is a Markov category, sometimes called the category of discrete probability. (Reference , example A.1.4) is the category of Tychonoff topological spaces and kernels which are valued in Radon probability measures, and where the measure varies continuous in with respect to the weak topology---in other words, given any continuous function the resulting function on given by is continuous. (A space is Tychonoff if it is Hausdorff and, given closed and not in , there exists continuous with for . Every locally compact Hausdorff space is Tychonoff). Eigil Fjeldgren Rischel 2025 4 27 https://erischel.com/efr-NKD7/ efr-NKD7 /efr-NKD7/ Diagram Markov Categories Example If is a Markov category and is any ordinary category, there is a Markov category whose objects are functors and whose morphisms are natural transformations between these considered as functors into (i.e natural transformations with stochastic components). The monoidal and Markov structure is defined simply component-wise. In particular, taking the walking arrow, we obtain a Markov category of deterministic arrows . We will denote this category simply . Again, the objects of this category are the deterministic morphisms of , while the morphisms are the commutative squares with not-necessarily-deterministic sides Eigil Fjeldgren Rischel 2025 4 27 https://erischel.com/efr-4VOU/ efr-4VOU /efr-4VOU/ A note on terminology Remark When we speak of a stochastic map in a Markov category, we always mean a map which is not necessarily deterministic---that is, a general map of . Given a map into a tensor product, the composites with the projections are called the marginals. Given two maps , we refer to any map with those marginals as a pairing of the two. Note that there is always a canonical pairing given by . We write for this canonical pairing. For deterministic maps this is just the usual pairing using the universal property of the product. (We mostly use this in cases where one map is deterministic, so that the pairing is unique assuming positivity). When , we say , and say the two coordinates are independent given (or just independent). Recall that the codomain functor is a fibration if and only if admits pullbacks. Since Markov categories have terminal objects but not, in general, products, they clearly cannot be expected to have pullbacks. However, if is positive (Reference , def. 11.22---but see Proposition below), given deterministic, every has a unique pairing with . Computing pullbacks in the subcategory we usually have a similar property. This will be the basic idea behind Markov prefibrations. By the following straightforward proposition, this property of having unique deterministic pairings is in fact equivalent to positivity: Eigil Fjeldgren Rischel 2025 4 26 https://erischel.com/efr-OYB6/ efr-OYB6 /efr-OYB6/ Characterization of positivity Proposition Let be a Markov category. The following are equivalent: In , if has deterministic marginal then . In other words, anything is independent of a deterministic variable. Given a deterministic morphism and any morphism there is a unique with those marginals. is positive. Eigil Fjeldgren Rischel2025426Proof Clearly 1 and 2 are equivalent, since independence just means is the independent pairing of the marginals---if it is uniquely determined by its marginals, it must be equal to the independent pairing, and conversely if it is necessarily independent, it is determined by its marginals. Now let us show this is equivalent to positivity. First suppose has unique pairings in this sense. Let be as in the definition of positivity. Then the two maps indicated are pairings of and , and since is deterministic, they are identical by hypothesis. Finally, the implication is precisely Reference , prop. 12.14. Eigil Fjeldgren Rischel 2025 3 27 https://erischel.com/efr-T3HM/ efr-T3HM /efr-T3HM/ Pullback-positive Definition Let be a Markov category. We say is pullback-positive if admits pullbacks and, given a diagram of this form where is deterministic, there is a unique map making the squares commute. Note that a pullback-positive category is in particular positive by taking . Eigil Fjeldgren Rischel 2025 9 14 https://erischel.com/efr-A7K7/ efr-A7K7 /efr-A7K7/ Remark The pullbacks appearing in Definition are of course pullbacks in , not , analogously to how is a product in , not . We will still use the notation for these pullbacks, which should not lead to any confusion. We may occasionally write instead of , if we are carrying out a construction which primarily involves . Since essentially no Markov categories have Cartesian products (except when the tensor product is Cartesian), this should also not lead to any ambiguity. Eigil Fjeldgren Rischel 2025 3 27 https://erischel.com/efr-0YP1/ efr-0YP1 /efr-0YP1/ Lemma Suppose any category admits products and intersections---that is, pullbacks whenever are subobjects of . Then it admits all finite limits. Suppose admits and preserves pullbacks along monomorphisms. Suppose further is positive. Then it is pullback-positive. Eigil Fjeldgren Rischel2025327Proof To see the first point, let be arbitrary maps. Note that , where the horizontal map is given by and the top by ---in the sense that the universal property of this intersection is exactly the universal property of the given pullback. Now suppose preserves this intersection, and is positive. (Note that it doesn't follow that the inclusion preserves the pullback , because it doesn't preserve the products). This amounts to the claim that a map lifts to the pullback if and only if the composite lifts over the map (note that the pullback of a monomorphism is a monomorphism). This implies in particular such a lift is always unique. Since if is positive, given , if the latter is deterministic there is a unique pairing this implies there is at most one map pairing the two. On the other hand, since the composite is deterministic and equal to the composite , they are both deterministic, and hence the pairing factors over the diagonal. Applying positivity again, to the tensor product since the latter component is deterministic, this pairing is independent, and hence the map does indeed lift, finishing the proof. The idea here is that a distribution on a subobject defined by some condition is simply a distribution so that the condition is satisfied with probability . This is a natural condition which holds in many Markov categories. The assumption that is pullback-positive will play a key role in the development of the theory of Markov fibrations. Although the theory could possibly be developed without assuming the base category has deterministic pullbacks, positivity seems to be an essential part. Eigil Fjeldgren Rischel 2025 4 27 https://erischel.com/efr-MW29/ efr-MW29 /efr-MW29/ Lemma If is representable and positive, has limits, and the monad preserves intersections, then is pullback-positive. Eigil Fjeldgren Rischel2025427Proof This is clear, since if the monad preserves a given limit, so does the inclusion into the Kleisli category (for completely abstract reasons), hence by Lemma we are done. We will need the notion of support in a Markov category, introduced in Reference , for certain examples, so we briefly record the definition and a few of its properties here. Eigil Fjeldgren Rischel 2025 4 3 https://erischel.com/efr-AB57/ efr-AB57 /efr-AB57/ Support of a morphism Definition Let be two morphisms in a Markov category. We say is absolutely continuous with respect to and write if, whenever two maps are -almost surely equal, they are also -almost surely equal. Let be a morphism in a Markov category. The support of , if it exists, is an object which represents the functor of morphisms into which are absolutely continuous with respect to . Eigil Fjeldgren Rischel 2025 4 3 https://erischel.com/efr-8MYE/ efr-8MYE /efr-8MYE/ Proposition The support of is equipped with a canonical deterministic monomorphism , so that a map into is absolutely continuous with respect to if and only if it factors over the support. Two maps are -almost surely equal if and only if they are strictly equal on the support. Eigil Fjeldgren Rischel 2025 6 18 https://erischel.com/efr-NC0R/ efr-NC0R /efr-NC0R/ Double Categories Eigil Fjeldgren Rischel 2025 6 18 https://erischel.com/efr-RXP4/ efr-RXP4 /efr-RXP4/ Double Category Definition A (strict) double category is a category internal to the category of categories. Concretely, it consists of: A set of objects A collection of vertical morphisms forming a category with A collection of horizontal morphisms forming a category with An a collection of squares. Each square has a left and right boundary given by vertical morphisms , and top and bottom boundary given by horizontal morphisms so that and so on: The squares compose horizontally and vertically in the obvious way, each of which form a category (in particular, there are identity squares for each vertical and horizontal map). A double functor is a mapping on objects, vertical and horizontal morphisms, and squares, which preserves all the identities and composition. There is also a notion of pseudo double category, which weakens the horizontal composition to only be associative and unital up to a coherent system of squares. We will not go into the details here, see § for more on this. Eigil Fjeldgren Rischel 2025 6 18 https://erischel.com/efr-BEJ6/ efr-BEJ6 /efr-BEJ6/ Example There is a pseudo double category where the objects are categories, the vertical maps are functors, the horizontal maps are profunctors (functors sometimes called bimodules), and the squares are natural transformations. For any category with pullbacks , there is a pseudo double category with as the vertical category, spans as the horizontal morphisms, and commutative diagrams as the squares. There is a double category of sets, functions, and relations. For any Markov category , there is a double category with and . For any category at all, there is a double category with and commutative squares as the pullback squares. Eigil Fjeldgren Rischel 2025 6 18 https://erischel.com/efr-Y1S5/ efr-Y1S5 /efr-Y1S5/ Definition A double category is thin if, for each compatible square of vertical and horizontal morphisms, there is at most one square filling it. In other words, such a square either commutes or doesn't. Sometimes the two classes of morphism are instead called loose and tight, especially in cases where the composition of the loose class is not associative, or if the loose class is a superset of the tight class. Although the explicit description above is probably the best way to think about the data of a double category, on a technical level it is often useful to think in terms of internal categories. An internal category in a category is a pair of objects , maps (the domain, codomain and identity), and a map (the multiplication), satisfying the usual laws of a category. A double category in the above sense is then an internal category in the category of categories. In the terms of Definition , is , and is the category whose objects are horizontal arrows, and whose morphisms are squares (composed vertically). Obviously, we could just as easily have oriented things horizontally, but this is the convention usually adopted. Given a double category , there is another double category called the transpose of , which has the same objects but exchanges the horizontal and vertical morphisms. In many cases it is not clear which of and is the "correct" one to work with, and we may have to pass back and forth between them. We try to stick to the convention that the horizontal morphisms are the "loose" ones. To avoid confusion, we will also simply specify the classes of morphisms directly, speaking for example of "the lenses" or "the charts" when working with the double category (Definition ). The concept of double category goes back to Reference . See Reference , section 12.3 for a textbook treatment. They have been widely used in applied category theory. An early application is in Reference , which constructed a "structured" version of the double category of cospans. Here the philosophy is that the horizontal morphisms are the "systems" (in a general sense) which are being wired together by horizontal composition, while the vertical morphisms (and squares) are "structure-preserving maps between systems". This is similar to the our double category of "bisystems" (§ ). Recent work by Lambert and Patterson Reference applies double categories to categorical algebra, with many applications to systems modeling, see Reference (as well as to classical category theory). In the next section we'll see their application to categorical systems theory, and later see them applied to parametrized morphisms. Eigil Fjeldgren Rischel 2024 6 30 https://erischel.com/efr-0023/ efr-0023 /efr-0023/ Review of Categorical Systems Theory We will give a review of the framework called categorical systems theory, due to Myers (Reference ). Eigil Fjeldgren Rischel 2024 6 6 https://erischel.com/efr-000A/ efr-000A /efr-000A/ Theory of Dynamical Systems Definition A theory of dynamical systems is an indexed category equipped with a section of its Grothendieck construction . It is called monoidal if is a monoidal category, and is a lax monoidal functor. It is further called symmetric monoidal if is a symmetric monoidal category and is a symmetric lax monoidal functor. Note that this is equivalent to requiring to be a (symmetric) monoidal fibration in the sense of Reference (see also Reference ). Eigil Fjeldgren Rischel 2024 6 30 https://erischel.com/efr-0024/ efr-0024 /efr-0024/ Lenses and charts Definition Let be an indexed category. A chart is simply a morphism in the Grothendieck construction . A lens is a morphism in the fiberwise opposite . Eigil Fjeldgren Rischel 2024 6 30 https://erischel.com/efr-0025/ efr-0025 /efr-0025/ Arenas Definition Given an indexed category , the double category of arenas has Objects the objects of ---note that these are the same as the objects of Vertical morphisms the lenses Horizontal morphisms the charts The double category is thin. Given a square of this form we can first project it to a square in . If this commutes, we can pull the lenses and charts back to a square in . The above square of lenses and charts will be said to commute if both of these squares commute. We will write arenas either as , or, for brevity when there is no need to discuss the two levels separately, simply with a symbol . We will also write both and as the situation calls for---there should be no confusion resulting from this. Eigil Fjeldgren Rischel 2024 6 30 https://erischel.com/efr-0028/ efr-0028 /efr-0028/ Dynamical system with interface Definition Let be an arena. A dynamical system with interface is an object and a lens . A morphism of systems is a map so that this square commutes in : The category of systems with interface is denoted Eigil Fjeldgren Rischel 2024 6 30 https://erischel.com/efr-0029/ efr-0029 /efr-0029/ Proposition Given a lens postcomposition gives a functor . Given a chart , there is a profunctor where the set over and is the set of maps so that this square commutes: This defines a double functor . The fibration encodes what sort of information can be "indexed over a space", while the section tells us what sort of information (such as a next step, or a gradient vector) must be produced to give a dynamical system. In this respect, the theory is very similar to the coalgebraic approach to dynamical systems, or systems that "do something", and indeed we have the following: Eigil Fjeldgren Rischel 2024 6 30 https://erischel.com/efr-0026/ efr-0026 /efr-0026/ Example Let be a functor. Then there is a dynamical systems theory where , is constant at , and . It is easy to see that the category of closed dynamical systems in this theory is equivalently the category of -coalgebras. Of course, coalgebras can already encode systems with input and output---the point of the CST framework is to separate out the input and output of systems so that they can be acted on in a compositional manner. Eigil Fjeldgren Rischel 2024 7 2 https://erischel.com/efr-002F/ efr-002F /efr-002F/ The theory of discrete dynamical systems Example There is a theory of dynamical systems with category of spaces , category of bundles (with pullbacks for reindexing), and tangent bundle . We call this the theory of discrete dynamical systems. (In the sense that they are both discrete-time and discrete-space). In another direction, we have the following comparison result: Eigil Fjeldgren Rischel 2024 6 30 https://erischel.com/efr-0027/ efr-0027 /efr-0027/ Example In the theory of discrete dynamical systems: The category of lenses is equivalently the category of polynomial functors and natural transformations between them. Using this equivalence, the category of dynamical systems with interface is exactly the category of -coalgebras and homomorphisms The first part here (which works, suitably formulated, for any locally Cartesian closed category), is a classical part of the theory of polynomial functors, see eg Reference . The second part is due to Spivak, see Reference . There is an extensive body of work on the description of interacting (discrete, deterministic) dynamical systems in terms of polynomial functors, see e.g. also Reference . Eigil Fjeldgren Rischel 2025 5 7 https://erischel.com/efr-GXJ0/ efr-GXJ0 /efr-GXJ0/ The theory of stochastic discrete dynamical systems Example There is a systems theory where (with the evident reindexing maps) and . In this theory, a closed dynamical system is a set equipped with a map . An open dynamical system has stochastic update function, but output which depends deterministically on the current system state. In fact this example works for any monad on the category of sets. Eigil Fjeldgren Rischel 2025 6 17 https://erischel.com/efr-KZEM/ efr-KZEM /efr-KZEM/ The theory of smooth dynamical systems Example There is a dynamical systems theory where is the category of smooth manifolds, is the category of fiber bundles over , with given by pullback along (note that pullbacks of bundles always exist), and with being the tangent bundle of in the ordinary sense. In this case, closed dynamical systems are manifolds equipped with (smooth) vector fields, which is what is classically thought of as a smooth dynamical system. Eigil Fjeldgren Rischel 2025 5 7 https://erischel.com/efr-9WQU/ efr-9WQU /efr-9WQU/ Clock system Example Consider the theory of discrete-time, discrete dynamical systems from Example . Let be a system. Consider the system given by in the forwards direction, and in the backwards direction. (To be clear, the object denoted is the map ---it is a singleton in each fiber). Then a morphism of systems consists of the following data: A function . Another function . A third function so that . So that and That is, it is a choice of a sequence of points in the state-space, and a sequence of inputs compatible with the outputs, so that this sequence obeys the dynamics. In other words, it is a trajectory of the system. Because of this, one views a generic chart map as a generalized trajectory, of a type given by the domain system. As another example, taking the state space to be with an update map that carries to one finds a system which classifies -periodic trajectories. Similarly, in the smooth case, the system classifies solutions of a smooth differential equation (those which extend to infinity). Recent work by Lynch, Myers, Staton, and the author (Reference ) constructs clock systems for theories of stochastic, discrete-time doctrines, although we will not delve into this here. Because this thesis is about Markov fibrations, we have chosen to present these ideas in terms of fibrations equipped with sections. Myers' book Reference actually prefers the presentation in terms of indexed categories. Similarly, we construct a double category of systems which is fibred (in a certain sense) over the double category of arenas. Myers instead displays this as a doubly indexed category , which carries lenses to functors and charts to profunctors. In a recent paper Reference , Myers and Libkind further develop the category theory of what they term double operadic categorical systems theory, which again concerns notions of "composable system" which are described in terms of such doubly indexed category (although for technical reasons, they use the language of right modules in that paper and a somewhat different presentation, the concept is the same.) Eigil Fjeldgren Rischel 2025 3 26 https://erischel.com/efr-O088/ efr-O088 /efr-O088/ Markov Fibrations and Stochastic Lenses Eigil Fjeldgren Rischel 2025 4 7 https://erischel.com/efr-X0ZV/ efr-X0ZV /efr-X0ZV/ Introduction Let denote the Kleisli category of the discrete (countable) distribution monad on . This is a simple setting for working with probability theory---sufficient for many applications. In order to study compositional Bayesian game theory (Reference , Reference ) one studies the category of optics in . These have the right level of expressivity to talk about players taking random actions, and where payoff depends stochastically on players' decisions. In . Games with this codomain naturally describe the situation of a player who has a binary choice between or . We call coproducts of this form the "good" coproducts---note that also , but this is considered somewhat pathological, since it relies on the nonexistence of any morphism when and are nonempty. In order to describe this structure on it would be useful if it had all coproducts. Unfortunately this is not the case. has a well-known extension with all coproducts, given by the fiberwise opposite of the fibration (this is Proposition in the case ). Extending this to stochastic maps would be the obvious way of constructing such a category of "dependent optics". Consider a category defined as follows. Its objects are the objects of ---that is, they are indexed families of sets. A map in is a stochastic map in the base , and a stochastic map on the total spaces which is compatible with it. Note that if is surjective, the map on the base is fully determined by the map on the fibers, which must merely satisfy the condition that the distribution of the indexing point in depends only on the indexing point in , not the specific point in the fiber . We claim is a reasonable notion of "stochastic charts". Recall that by "chart" we mean something like "lenses where both maps go forward". If stochastic lenses are supposed to include optics as the full subcategory spanned by the "non-dependent" objects, then the charts should include "co-optics"---that is, maps between and should be given by the coend And in fact this is the case: Clearly there is a map from this coend to maps in . By taking and conditioning on , we see this is surjective. Finally, by restricting to the support of the forwards part inside , we obtain a representative for each element of the coend which is uniquely determined by and (since the conditional is well-defined on the support). Note: This relies both on the fact that has conditionals, and on the existence of supports. We've previously seen these defined in abstract Markov categories (Definition )--supports in of a morphism are simply given by those so that 0]]>. Note that the existence of both conditionals and supports is a very strong assumption---the only categories we are aware of with both properties are those whose probability distributions have a discrete character, like and . Neither will be essential to the theory, but both will play a role in certain theorems---we will see more of this later. The goal of the theory of Markov fibrations is to give a notion of "fiberwise opposite" which can be applied to the codomain functor to give a reasonable notion of "stochastic lenses". In particular, we should recover the usual category of optics in the previous case. It is clear that the codomain functor is not a (Grothendieck) fibration, since this would require to have pullbacks, which can only hold for a Cartesian Markov category. However, we can do some things. Namely, given a Cartesian (pullback) square in and a map where the base map is deterministic, for each deterministic factorizing map there is a unique lift . In other words, the pullback over is a fibration---in fact, it is simply the family fibration . Furthermore, if is itself deterministic, there is such a unique lift even without assuming that the factorization is deterministic. Moreover, we can factor any map in as such an induced map followed by a map over a deterministic base, as follows: This gives us a hope that we can, in some way, control the category using the pullback over , which is a fibration, and some information somehow given by these extra maps. Note also that the diagram above is equivalent to giving: a span and a section , which all lives in the base, and a map in the fiber over . Thus it would seem to be very amenable to fiberwise dualization. Analogous to our argument above that "co-optics" are equivalent to maps in we can do the following: Suppose given two tuples as above. Suppose there exists a map over , so that . Then there is a canonical map over , because pullbacks commute. If the triangle moreover commutes, then these two triples represent the same map in These equivalence relations correspond to "sliding" for deterministic morphisms . Note that the condition here can be checked just on the fibration . To obtain the full set of sliding equations, we will need to use stochastic maps , and thus leave that fibration behind. However, we are tantalizingly close to realizing as being presented by some sort of additional structure on the fibration . (For a general monad acting on , the category , of effectful optics up to sliding of pure morphisms, was studied by Riley in Reference , section 4.9, and by Hedges in Reference ) In this chapter we will indeed provide such a structure, and analyze it. In § , we'll axiomatise the lifting property of discussed above into a property we call a Markov prefibration (Definition ). In § , we exhibit a free Markov prefibration associated to a fibration---its morphisms are precisely spans of the form seen above. Naturally, given a prefibration , its underlying fibration on becomes an algebra for the monad of this adjunction. In § , we characterize the class of prefibrations which are presented by their underlying algebra in this way---these are the Markov fibrations (Definition ). Since the monad commutes with fiberwise opposites, this yields a notion of fiberwise opposite for Markov fibrations. Following this, we review a few properties of the theory of Markov fibrations, including the existence of coproducts in the fibration (Proposition ), the stability of Markov fibrations under limits (§ ), and induced monoidal structures (§ ). Combining these, we can prove: Eigil Fjeldgren Rischel 2025 4 27 https://erischel.com/efr-OSH4/ efr-OSH4 /efr-OSH4/ Theorem There exists a (strong) symmetric monoidal functor which Is fully faithful. Preserves the good coproducts. So that the image has all coproducts, and every object of the image is a coproduct of objects in And so that moreover the monoidal structure on the image is distributive Eigil Fjeldgren Rischel 2025 3 26 https://erischel.com/efr-2IMZ/ efr-2IMZ /efr-2IMZ/ Markov Prefibrations Eigil Fjeldgren Rischel 2024 11 21 https://erischel.com/efr-0045/ efr-0045 /efr-0045/ Remark Markov categories generally do not have pullbacks, for the same reason that they usually don't have products. This issue generally hinders the construction of fibrations, in the ordinary sense, of Markov categories. However, we can go part of the way. The idea of the following definitions is that given a pullback in the deterministic category, say , a map where the -coordinate is deterministic should be uniquely determined by a choice of (deterministic) map and (stochastic) such that the square commutes---as we claimed above (and will see below), this holds for the Markov category of discrete probability . The analogous statement for products---that a map with deterministic -component is uniquely determined by the projections (or marginals) ---is a consequence of positivity (see Proposition ), and hence holds in most Markov categories of interest. Eigil Fjeldgren Rischel 2024 6 24 https://erischel.com/efr-0019/ efr-0019 /efr-0019/ Markov prefibration Definition Let be a Markov category, and let be a functor into it. Then we call a Markov prefibration if the following two conditions hold: The pullback is a (Grothendieck) fibration Given maps , in , such that are deterministic and are Cartesian for the above fibration, induces a bijection between maps such that , and maps so that . Note that when restricted to those maps where is deterministic, this being a bijection is the defining property of being Cartesian (for any , not necessarily a Cartesian one). Given a Markov prefibration we write for . We will refer to this as the deterministic part of ---note that this does have the potential for confusion, as when is itself a Markov category, this is not necessarily the same as the deterministic subcategory of . When lies inside , and is Cartesian for that fibration, we will simply refer to it as a Cartesian map in (there are no other types of Cartesian maps, so this should not lead to confusion). A morphism of Markov prefibrations is a functor over which preserves Cartesian maps. The category of Markov prefibrations over thus defined is denoted . Taking the deterministic part defines a functor . Eigil Fjeldgren Rischel 2025 4 6 https://erischel.com/efr-EQ41/ efr-EQ41 /efr-EQ41/ On 2-category theory Remark Since we will shortly be working with a number of functors between categories whose objects are themselves categories with some structure, it may be thought that we should give some consideration to the strictness of our constructions---for example, we will shortly construct a left adjoint to and it may well be asked how strict this adjoint is, whether we need to consider the definition of pseudomonad when we get so far, et cetera. However, we can largely avoid this issue. The key observation is that none of our functors will alter the objects of the underlying category (since is identity on objects,). Hence, all the natural transformations that we would ordinarily ask to be equivalences of categories will instead be isomorphisms, and we can largely ignore considerations of higher category theory---similarly, all our functors will be strictly functorial. As a simple example of this, observe that the pullback functor is automatically strict---it simply consists in restriction to a subset of the morphisms in (which is automatically closed under composition), and thus clearly preserves composition strictly. Eigil Fjeldgren Rischel 2025 4 29 https://erischel.com/efr-DINO/ efr-DINO /efr-DINO/ Definition In a Markov prefibration (as previously noted), a morphism in is called Cartesian if is deterministic and is Cartesian in the fibration . It is called vertical if is an identity. It is called a stochastic-Cartesian if there exists a Cartesian map so that (recall that in this case is uniquely determined by and ). Note that if is stochastic-Cartesian and is deterministic, then is Cartesian. The introduction to this chapter contains the argument that is a Markov prefibration. This is a key motivating example. Eigil Fjeldgren Rischel 2025 4 8 https://erischel.com/efr-CJTH/ efr-CJTH /efr-CJTH/ Markov prefibrations over Cartesian base Example Let be a Markov category which is Cartesian (that is, one where all morphisms are deterministic). Then a Markov prefibration over is simply a Grothendieck fibration. Eigil Fjeldgren Rischel 2024 11 19 https://erischel.com/efr-0044/ efr-0044 /efr-0044/ Proposition Let be a Markov category. Then is a Markov prefibration with Cartesian maps given by pullback squares, if and only if is pullback-positive. Eigil Fjeldgren Rischel20241119Proof Assume first is pullback-positive. It is clear that computing pullbacks in gives the required Cartesian lifts---pullback positivity is precisely the claim that lifts exist uniquely in the definition of a Cartesian morphism (since the map to the base leg of the pullback is always deterministic in this case). Since Cartesian lifts can be taken to be deterministic (we have just constructed deterministic Cartesian lifts, and such lifts are unique up to unique isomorphism), the second condition also follows from this assumption, simply taking the other leg to be deterministic. Conversely, suppose is a Markov prefibration and suppose the Cartesian maps are given by the deterministic pullback squares. Let be an object of and let be a deterministic map, and form the pullback , which is Cartesian. Let be deterministic and let be any map. Expanding the latter into a map from the object to the Cartesian property of the square implies there is a unique pairing Eigil Fjeldgren Rischel 2024 11 19 https://erischel.com/efr-0040/ efr-0040 /efr-0040/ Remark In a general Markov category, not every isomorphism is necessarily deterministic. This means that, in general, fibres over isomorphic objects in a Markov prefibration are not necessarily isomorphic or even equivalent as categories. This seemingly immoral situation is, in fact, in accordance with other results indicating that deterministic isomorphism is really the proper notion of identification in a Markov category. See eg Reference , Section 4, for further discussion of this point. (Since the basic idea of a Markov category involves objects equipped with some structure which is not preserved by all the morphisms, it is not so paradoxical that an isomorphism in this situation should be insufficient to render two objects identical). Under very weak assumptions on the Markov category , such as positivity, all isomorphisms are deterministic. This implies that all Markov prefibrations over are isofibrations, and thus rules out any sort of behavior like the above. As noted, we are only concerned with positive Markov categories. Relatedly, in the proof of Proposition , a careless prover may have erroneously concluded after the first step that all Cartesian lifts are deterministic---but since Cartesian lifts are characterized only up to isomorphism, not necessarily deterministic isomorphism, this does not automatically follow. (But of course replacing a nondeterministic lift with an isomorphic deterministic one in this situation cannot alter the unique existence of the factorization, so it does not matter). Eigil Fjeldgren Rischel 2024 11 19 https://erischel.com/efr-0042/ efr-0042 /efr-0042/ Proposition Let be a Markov prefibration, let be full subcategories so that is contained in , and suppose is a monoidal subcategory (which is then automatically a sub-Markov category). Suppose finally is stable under pullback along deterministic morphisms in . Then is again a Markov prefibration. Eigil Fjeldgren Rischel20241119Proof By assumption, given and , the Cartesian lift is again in . The fullness of the subcategory inclusions suffices to prove the existence and uniqueness of the required lifts so that this is still Cartesian after restricting. For the same reason, since we have just observed that the Cartesian lifts are the same as in , the second part of the definition also holds. Eigil Fjeldgren Rischel 2025 3 27 https://erischel.com/efr-NC7D/ efr-NC7D /efr-NC7D/ as Markov prefibration Example Recall that the deterministic maps in are those kernels valued in -valued measures. (And that these are not the same thing as the measurable maps). Given a pair of such maps, we claim that the subset of given by those points where as measures on , equipped with the subset -algebra and its inclusions into and , is a pullback in . To see this, observe that is the Kleisli category for the submonad of the Giry monad which takes a space to the subspace of -valued probability measures on it. Note that by the general theory of Markov categories this has products given by the products in . It hence suffices to prove that the pullback of along the diagonal exists---if it does, it has the universal property of the product . To see that it does (and that it's given by the space described above), note that the diagonal is a split monomorphism, so it suffices to show that factors over if and only if the composite to lifts over the diagonal. Now this lift over exists if and only if the probability of the diagonal under the composite is . By definition, it is Since the function being integrated is an indicator, this is simply the measure of the set If the probability of the diagonal is one, clearly the two marginals agree. Conversely, since is Cartesian, it must be the case that if the marginals agree the probability of the diagonal is 1. Therefore this is equal to the subset described at the start. But for this to have measure for all is equivalent to the desired lifting property. This finishes the argument. Moreover, this argument relied only on the -valuedness of the maps , not or its marginals. Hence this also proves that the pullback along the diagonal is preserved by the inclusion . Since is known to be positive, this implies is a Markov prefibration. Eigil Fjeldgren Rischel 2024 11 19 https://erischel.com/efr-0041/ efr-0041 /efr-0041/ Proposition The codomain functor is a Markov prefibration. Eigil Fjeldgren Rischel20241119Proof We wish to apply Proposition . The only thing to check is that standard Borel spaces are stable under pullbacks in . But standard Borel spaces are known to be stable under products and measurable subsets, and this is enough (see eg. Reference propositions 3.1.23 and 3.3.15) Eigil Fjeldgren Rischel 2024 11 19 https://erischel.com/efr-0043/ efr-0043 /efr-0043/ Corollary is a Markov prefibration. Eigil Fjeldgren Rischel 2025 3 29 https://erischel.com/efr-8B5X/ efr-8B5X /efr-8B5X/ Example The functor is not a Markov prefibration, although its pullback over is a Grothendieck fibration. To see this, first consider the deterministic pullback. An optic with deterministic base can be identified with a map (and the base deterministic map ). To see this, first observe that the subset of with the marginal deterministic is in bijection with , since is positive. Hence we can calculate using the ninja yoneda lemma as in Proposition . Hence this part is a fibration with the fiber over being the coKleisli category of the monad, and the pullback functors given by reindexing these parametrized maps. The Cartesian lift of a map at is given by the optic with unit residual and identity backwards component. Now, let denote the standard Gaussian distribution, let be the function given by if and otherwise, and consider the two optics given by (where we recall that the first argument is the residual). They cannot be equal, as postcomposition with the optic given by the identity yields, for the former, the standard Gaussian , and for the latter, the constant zero map. But postcomposition with the projection does give the same optic (the identity), because, for every fixed , when is normally distributed, with probability one. Hence the unique lifting of Cartesian maps over Cartesian maps cannot hold. This counterexample indicates that, although is a Markov prefibration, we can not expect a dual version of this prefibration---in fact, since over deterministic maps is the fiberwise dual of (the restriction to trivially-indexed objects of) this example shows that there is no Markov prefibration whose deterministic part is the fiberwise dual of . Moreover, as the example indicates, this is not a mere technical issue, but an unavoidable fact about optics in general measurable spaces---even up to behavioral equivalence, they simply don't satisfy the conditions of being a Markov prefibration. (But see Theorem ) On the other hand, we have: Eigil Fjeldgren Rischel 2025 4 3 https://erischel.com/efr-UDS9/ efr-UDS9 /efr-UDS9/ Proposition Let be a positive Markov category with supports. Then is a Markov prefibration. Eigil Fjeldgren Rischel202543Proof As above, we see that the deterministic part is a fibration, so take deterministic maps, and let be an optic so that the induced triangle with the two Cartesian lifts to commutes. Let the two parts be . The implication is that -almost surely, is equal to the projection to (and renders the triangle in commutative). Note that factors over the support of this map, hence we can assume the marginal has full support. Hence up to sliding equivalence, is strictly equal to the projection. This implies the lift is uniquely determined as desired. The existence of supports rules out the pathological behaviour. Essentially, in the presence of supports, we can sensibly reason about "the points of measure zero" and exclude them from consideration---and Proposition implies that the independent pairing of two measures always have the least "points of measure zero", and so that what can be proven equivalent under the assumption of independence will always be equivalent. By contrast, the map from Example satisfies for almost all when is normally distributed, for all , but this does not imply that for all measures on with this marginal, is distributed as the marginal of . One point of view is that the map is simply pathological, and we should restrict our attention to maps that are continuous in some sense (from the point of view of computer science, one argument for this is that computable maps are necessarily continuous). The category of Tychonoff spaces and weakly continuous kernels does indeed have supports. However, since it lacks conditionals, it is still not ideal from our point of view. Eigil Fjeldgren Rischel 2025 3 26 https://erischel.com/efr-GO6R/ efr-GO6R /efr-GO6R/ Free Markov Prefibrations Because every map in factors into arrows which are "induced" from arrows in and the Markov prefibration property, it may initially be hoped that is in some sense "free" on the data of the fibration and the inclusion . If that was true, we may further hope that taking the fiberwise opposite of the fibration and applying the same free generation principle would generate a good notion of stochastic lens. Unfortunately, this is not the case. We will see that the free prefibration is given by gadgets which look a bit like an indexed version of optics, up to a sliding equivalence for deterministic maps on the residual. This prompts us to look for some extra structure on the fibration which describes sliding equivalences for stochastic maps on the residual. In the next section, we will see that this is exactly the structure of an Eilenberg-Moore algebra for the free prefibration monad on . In this section, we will give a description of the free Markov prefibration on a fibration (assuming is pullback positive). There is a fairly simple description of the hom-sets, but their composition is a bit tricky, and verifying associativity even more so. Hence we will employ a technical trick: by characterizing the hom-sets as "freely generated" in a certain sense from the hom-sets in , we can identify them with sets of natural transformations using a Yoneda-type argument, and infer composition and associativity from there. Eigil Fjeldgren Rischel 2024 6 25 https://erischel.com/efr-001I/ efr-001I /efr-001I/ Indexed copresheaf Definition Let be a functor. An indexed copresheaf on is a tuple consisting of a copresheaf, an object of , and a natural transformation as indicated. We say the indexed copresheaf is over , and we will abuse the terminology by referring to itself as an indexed copresheaf, leaving the transformation implicit (for example, "let be an indexed copresheaf over "). We denote the subset for by . A map of indexed copresheaves is a natural transformation and a map so that the obvious square of natural transformations commutes. We denote the category of indexed copresheaves by . Note that there is an obvious forgetful functor Observe that for each object , there is a corepresentable copresheaf . Maps between these obey the Yoneda lemma, in the sense that they are in bijection with maps between the underlying objects in . This defines a fully faithful functor over . Of course, there is a dual notion of indexed presheaf, but this will not interest us. Eigil Fjeldgren Rischel 2025 3 27 https://erischel.com/efr-FOJT/ efr-FOJT /efr-FOJT/ Proposition Let be any functor and let be an identity-on-objects functor. Write for the pullback. If is a copresheaf indexed over , the pullback is a copresheaf on indexed over again in a unique way. This defines a functor . Moreover, this functor preserves the corepresentable copresheaves (since is identity on objects, so is , so this statement makes sense). Eigil Fjeldgren Rischel 2025 3 27 https://erischel.com/efr-FRQG/ efr-FRQG /efr-FRQG/ Corollary Let be a functor and let be identity-on-objects and faithful. Then the pullback of the composite and the inclusion is identical to . Therefore, Proposition gives a functor . The idea of our construction of the free Markov prefibration is to give a certain monad on and consider the Kleisli maps between the representable copresheaves. At this point, the notion of a deterministic map equipped with a stochastic (ie not necessarily deterministic) section begins playing a key role. The phrase "stochastic section" will always carry an implicit "of a deterministic map". In most cases the map that the section is a section of will be clear from the context. Note that, given a stochastic section and a deterministic map , if is pullback-positive, there is a unique lifting of this to a section of the projection . We will use this fact several times. Eigil Fjeldgren Rischel 2025 3 27 https://erischel.com/efr-Y926/ efr-Y926 /efr-Y926/ Stochastic Module Definition Let be a Markov category, and let be a fibration. Then a stochastic module consists of An indexed copresheaf For each , Cartesian morphism lying over , and stochastic section , a function , which acts on the underlying morphisms in as composition with Satisfying, whenever given a commutative square of Cartesian morphisms: and stochastic sections lying over so that we have a digram in : Where the maps except are deterministic, and are sections of and , and both the square of deterministic maps and the square involving commute, the condition that the square commutes. And satisfying furthermore the equation, for every two stochastic sections the equation (note that this makes sense because pullbacks compose). A morphism of stochastic modules is an indexed natural transformation which preserves the operations . The category of stochastic modules is denoted . There is an apparent forgetful functor . Eigil Fjeldgren Rischel 2025 4 27 https://erischel.com/efr-GL6R/ efr-GL6R /efr-GL6R/ Remark Note that of course depends on , not just . If is a deterministic map, is a stochastic section, and are two Cartesian lifts of to , then applying the commutativity axiom for stochastic modules implies that the triangle commutes. In what follows, we will simply write for some choice of cartesian lift, and speak of . The above shows that this is a harmless abuse---the actions are preserved by the identification of different Cartesian lifts. In particular, we will often make arguments as if pullbacks compose strictly, although in general they only compose up to isomorphism. The above triangle means this is harmless. The term "stochastic module" is not very good, but this is mostly a nonce definition in any case, so we won't worry too much about it. Stochastic modules over a given can be seen to be monadic over the category of indexed copresheaves over that . However, the compatibility of these local left adjoints with the structure of the rest of the category is somewhat subtle. However, we do have free stochastic modules on representable indexed copresheaves, as we will soon see. Eigil Fjeldgren Rischel 2025 3 27 https://erischel.com/efr-2XPE/ efr-2XPE /efr-2XPE/ Proposition Let be a Markov prefibration, and let . Then the corepresentable copresheaf , restricted to , (but not pulled back---that is, we remember the whole set , even the part over stochastic , but only the composition with maps in ) is a stochastic module in a canonical way, with given by composition with the unique induced lift of . Moreover, any morphism of Markov prefibrations induces a homomorphism of stochastic modules Eigil Fjeldgren Rischel2025327Proof Given and a stochastic section , it's clear that composition with the unique lift is a map of the right type, so we just have to verify the equations. For the first equation (item 3 in the definition of stochastic module), we are comparing two maps . These are given by composition with two maps, let's call them . These two maps are lifts of and , but by assumption these two are equal. Hence by the uniqueness property of Markov prefibrations, , and we have our equation. The other equation follows in a completely analogous way. To prove the homomorphism property, note that a morphism of prefibrations preserves Cartesian morphisms, and hence (by uniqueness) must preserve the unique lifts of stochastic sections. Then by functoriality it must preserve composition with these, which finishes the proof. Eigil Fjeldgren Rischel 2025 3 27 https://erischel.com/efr-OH7U/ efr-OH7U /efr-OH7U/ Lemma Let be an object. Then there is a free stochastic module on its representable copresheaf, in the sense that if is another stochastic module, homomorphisms are in bijection with indexed natural transformations The free stochastic module on a corepresentable presheaf is given as follows: an element of consists of a diagram in of the form (where ,) where and are deterministic, plus a map lying over . This is up to the equivalence relation which, given some other such tuple, identifies them whenever there exists deterministic as in this diagram: so that the two deterministic triangles commute, and so that the unique Cartesian map over forms a commutative triangle with the two maps to . (Note that we do not claim the relation just described is inherently an equivalence relation, rather we form the equivalence relation generated by this). It is clear how a map acts on this to make it a copresheaf. It is indexed by taking a tuple as above to the composite . Given with a stochastic section , and an element of the induced element in is given by forming the pullback taking the pullback of the map over to one lying over the projection and composing the section with the induced lift The underlying copresheaf of this respects Cartesian maps in , in the sense that given a Cartesian map and an element lying over a deterministic map the natural map from lifts to lifts is a bijection. Eigil Fjeldgren Rischel2025327Proof First, we have to verify the equations of a stochastic module for . For the first case, suppose we are given a square with all but the upwards maps stochastic. Now let be some object over and suppose we are given an element of , represented by a span , a section and a map over . Consider then the below diagram: By definition, the first of the two possible elements of are given by either forming the pullback , taking the lift of to and composing to get a section , and pulling back along the projection to get a map , then taking the span . The second is given by first composing with the map , then applying the above procedure with the pullback . The induced map exhibits the equality of these two under the equivalence relation defining (commutativity of the bottom-right square implies that triangle of stochastic sections commutes.) Given some other stochastic module over with a map of indexed copresheaves over , there is at most one extension to a map of stochastic presheaves over ---given an element with representative it must go to the identity element of acted on by the stochastic section to produce an element of followed by applied to the map over . But it is not hard to see that the equivalence relation imposed by is implied by the equations of a stochastic module, and so this map is well-defined, establishing the property. Secondly, let be a Cartesian map over and take a commutative triangle of deterministic maps. Finally take an element of over the given map . We must show it has a unique lift to over the given map . Let us take a representative given by a diagram: First, note that the two maps and do not necessarily agree. However, we can remedy this by replacing by their equalizer---note that as we argued above, this gives an equivalent element of the stochastic module. (By writing their equalizer in as the split pullback we can see that the section factors over this, even if does not have all equalizers in general). Hence we can assume the triangle formed by adding the dashed arrow commutes. Since now factors over , the lift gives a lift . Now by the Cartesian property of , there is a unique lift of to this map (here we just use the fact that is a fibration). This gives the desired lift. Finally, given two distinct lifts (again, we can assume their maps factor over ,) clearly any map witnessing an identity between their composites would likewise exhibit an identity between their lifts (since pullbacks compose). This proves uniqueness, and finishes the proof. We are now ready to prove the main proposition of this section: Eigil Fjeldgren Rischel 2025 3 27 https://erischel.com/efr-LGOA/ efr-LGOA /efr-LGOA/ Proposition Let be a pullback-positive Markov category and let be a fibration. Consider the full subcategory of stochastic modules spanned by the free modules on the corepresentables. Denote the opposite of this category . Clearly there is a commutative diagram We claim: is a Markov prefibration. There is a bijection . When the left-hand side is equipped with the canonical stochastic module structure, and the right is equipped with the free one, this is moreover a homomorphism (hence isomorphism) of stochastic modules. is initial among Markov prefibrations receiving a map from . In other words, this construction gives a left adjoint to the pullback functor Eigil Fjeldgren Rischel 2025 3 27 https://erischel.com/efr-0BES/ efr-0BES /efr-0BES/ Proof First observe that, by Lemma , the pullback is indeed a fibration, with the image of the Cartesian lifts under the functor being Cartesian again. (This also establishes that really does receive a map of fibrations from ) Given Cartesian , and a stochastic lift consider the pullback , and the pullback of to it. There is a unique lift of to a section , and this induces a unique lift using the stochastic module structure. The composite of this with the projection to is a lift of over , as required by a Markov prefibration. Analogously to the proof of Lemma , given some other lift , the fact this is a factorization implies the existence of some with maps and a lift of the section , so that the induced map between the pullbacks over and of makes the triangle into commute. But then since this is a triangle over deterministic bases, this implies the lifted triangle to also commutes, hence this lifts to another representative of the lift we started with. But then it's not hard to see that this maps to and exhibits an equation with the previously constructed "canonical" lift. Hence is a Markov prefibration, and by the above, the induced stochastic module structure on the corepresentable presheaves is exactly the one given by (in other words this equation is not merely a bijection of sets, but an isomorphism of stochastic modules). Let be a functor over into some other Markov prefibration which preserves Cartesian maps. Using the algebra structure on we see there is a unique extension of to which respects the stochastic module structure. By chasing the diagram around it's easy to see that this is functorial, and hence gives a map of Markov prefibrations---conversely, any such map extending must be a stochastic module homomorphism. Thus there is a unique functor, proving initiality. This characterization of the left adjoint makes it fairly easy to understand the induced monad on . Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-ACIV/ efr-ACIV /efr-ACIV/ Remark We will sometimes refer to the morphisms of as precharts. Taking the fibration as an example, it is not too hard to see that the precharts between and are representatives of co-optics . (To see this, note that any prechart is equivalent to one where the apex of the span has the form and the right leg is the projection to . Then the rest of the data is a map and a map , since the -coordiante of the latter map is determined by the span). In fact their equivalence relation is given by sliding equivalence for deterministic maps (i.e morphisms in ). The precharts in will be called prelenses. We will speak of the tuple representing a prechart just as a "decorated span (representing ...)". When part of the structure is understood, or can just be left abstracted, we will denote such a decorated span simply by or even just . It will be clear from context which part of the structure is being specified. Eigil Fjeldgren Rischel 2025 4 7 https://erischel.com/efr-A08L/ efr-A08L /efr-A08L/ Proposition Let be a fibration. Then the underlying fibration of the free Markov prefibration, has fiber over given by Objects are simply objects of A morphism consists of a deterministic , a stochastic section and a map , up to the equivalence relation generated by, whenever is deterministic and is a factorization of , identifying with . Given two such morphisms , their composite is represented by equipped with the section formed as the composite of and the lift of to the pullback, and the composite Given deterministic , the pullback is given on such a map by taking the pullback the induced section, and the pullback of the map along the projection It is immediately obvious that we have: Eigil Fjeldgren Rischel 2025 4 7 https://erischel.com/efr-ILIQ/ efr-ILIQ /efr-ILIQ/ Corollary The free Markov prefibration monad commutes with fiberwise opposites. In particular, algebra structures on are in bijection with algebra structures on and fiberwise opposites lifts to an involution of . Eigil Fjeldgren Rischel 2025 4 25 https://erischel.com/efr-OEFJ/ efr-OEFJ /efr-OEFJ/ Stochastic Module Fibration Definition Let be a pullback-positive Markov category. The adjunction induces a monad on . A module for this monad is called a stochastic module over (or, to distinguish it from the copresheaves of Definition , a stochastic module fibration). The category of stochastic module fibrations is denoted Let us try to understand the structure of a stochastic module fibration. It is easiest to understand in the case of a projection map . Suppose we have two objects over . Then we think of a map over as a map , that is a map parameterized by (this is literally the case for a codomain fibration). Given a stochastic section of , which amounts to a stochastic map , the stochastic module stucture picks out a new map , which is given over each point by choosing the parameter according to , then applying . The definition of the composite in Proposition , in these terms, tells us that given maps , and maps , the composite of and is equal to the map obtained by forming the parameterized composite and applying the independent pairing . This is of course how composition is supposed to work in a Markov category. Eigil Fjeldgren Rischel 2025 4 25 https://erischel.com/efr-VF6V/ efr-VF6V /efr-VF6V/ Lemma Let be a stochastic module over , and let be given, so that every map except is deterministic. Suppose the deterministic part of the diagram commutes, , is the induced section, and is a section. Let be two objects over . Suppose given a map . Then In particular, this operation depends only on . Moreover, it is functorial, in the sense that given a diagram with the downwards maps deterministic, Eigil Fjeldgren Rischel 2025 4 25 https://erischel.com/efr-U436/ efr-U436 /efr-U436/ Remark The operations associated to stochastic lifts are "functorial" in the sense that . However they are not functorial in the sense that To make sense of this, consider a simple case of a map in . Given two objects over (in ), a map is equivalent to a parametrized map . The operation consists in sampling this parameter according to the distribution ---but since composition in the fiber over is defined by copying the parameter, but composition in the fiber over (i.e just ) is defined by composing the kernels under conditional independence, these only agree if the distribution is assumed to be deterministic. Note that if either or is pulled back from a map (i.e, if they do not depend on the parameter ), the composition is preserved. By construction, two morphisms in represented by spans with apex are identified if there exists a zig-zag of spans (decorated with sections from the domain and morphisms in the fiber, satisfying equations, etc). We will now prove a lemma that allows us to cut this down to a smaller set in many conditions. We will need the following hypothesis: Eigil Fjeldgren Rischel 2025 4 15 https://erischel.com/efr-2MH5/ efr-2MH5 /efr-2MH5/ Weak conditionals Definition Let be a Markov prefibration. We say (or, abusing notation, ) admits weak conditionals if, given a Cartesian map and any map the existence part of the Cartesian condition holds---that is, every factorization admits a lift, although not necessarily a unique one. We say a Markov category admits weak conditionals if its codomain functor is a prefibration which admits weak conditionals---this is equivalent to requiring that it is pullback-positive, and that all deterministic pullbacks are carried to weak pullbacks by the inclusion (in other words, that they satisfy the existence part of the universal property even for pairs of nondeterministic maps). Observe that, if admits conditionals, it certainly admits weak conditionals: given a pullback and maps form a Bayesian inverse of with respect to the given measure, and use that to build a lifting , which gives ---then a diagram chase verifies that this map has the desired properties. Eigil Fjeldgren Rischel 2025 4 11 https://erischel.com/efr-TZ3Y/ efr-TZ3Y /efr-TZ3Y/ Lemma Suppose admits weak conditionals, and let be a fibration. Then two morphisms in , represented by commutative diagrams as well as , for , are equal if and only if there exists a span over , with a stochastic section lifting both the sections to , so that the pullbacks of to agree. Eigil Fjeldgren Rischel2025411Proof The relation here described clearly implies identity, and contains all the generating identities, so it suffices to show it is an equivalence relation. Reflexivity and symmetry are clear, so transitivity is the only issue. It suffices to show that, given a span and maps so that the triangles commute, and so that the two induced sections agree, and a map which pulls back to , we can find as above. To do this, take . Clearly the maps to are over , and by the existence of weak conditionals there exists a common lift of the sections to . By functoriality of pullbacks, the pullbacks of to agree. Eigil Fjeldgren Rischel 2025 3 29 https://erischel.com/efr-U4RA/ efr-U4RA /efr-U4RA/ Markov Fibrations Recall that, given a functor with left adjoint , there is a "standard resolution" of any object , given by the "cofork" , where the two parallel maps are the two possible applications of the adjunction counit. The adjunction is monadic if and only if this is always a coequalizer, in which case the -algebra corresponding to is ---conversely, given an algebra , there are two parallel maps (given by and the counit,) and the object in corresponding to this algebra is given by this coequalizer. Eigil Fjeldgren Rischel 2025 3 29 Example Consider the adjunction between the category of monoids and the category of sets. Given a monoid , consists of lists of elements in , and consists of lists of such lists. The two maps consist in either concatenating the lists, or replacing each list with its product. Clearly these two maps are coequalized by the product map . Moreover it's clear that two lists have the same product if and only if they are identified in this coequalizer (simply consider a singleton list-of-lists, which identifies any given list with the singleton corresponding to its product). As a generalization of this, if this coequalizer exists for every algebra, they form a left adjoint to the canonical functor . Since we have seen that the monad of free Markov prefibrations commutes with taking fiberwise opposites, we may hope that such a left adjoint exists---a simple argument shows that, if it is, it is fully faithful, and we may say that those prefibrations in the image are the "fibrations" and define their fiberwise opposite as the fiberwise opposite applied to their underlying algebras. Although it turns out to not be quite so simple, we will take this idea as our starting point. Eigil Fjeldgren Rischel 2025 3 29 Example Let be the category of topological groups, and let forget both the group structure and the topology. Clearly this is right adjoint to the free group in the discrete topology, and the monad of this adjunction is the free group monad, which we write for now. The canonical comparison functor just forgets the topology. Given a (non-topological) group, described by a map , we can form the diagram of topological groups . Here is the free, discrete group on and is the free discrete group on the underlying set of . Their coequalizer is simply equipped with the discrete topology, which is indeed the left adjoint to In what follows, we will denote the monad simply by to avoid too many complicated nestings of overlines and parentheses. Eigil Fjeldgren Rischel 2025 3 27 https://erischel.com/efr-HVUT/ efr-HVUT /efr-HVUT/ Markov Fibration Definition If is a Markov prefibration, we call it a Markov fibration if the diagram is a coequalizer in . Given an algebra of the free Markov prefibration monad, let be as above. We say presents a Markov fibration if the coequalizer of these maps in is a Markov prefibration. The terminology "presents a Markov fibration" is justified by the following proposition. Eigil Fjeldgren Rischel 2025 3 27 https://erischel.com/efr-MAUM/ efr-MAUM /efr-MAUM/ Proposition Let be stochastic module which presents a Markov fibration. Then the underlying algebra of the Markov prefibration obtained as the coequalizer of is isomorphic to , and in particular this prefibration is a Markov fibration. This correspondence determines an equivalence of categories between the full subcategory of spanned by the Markov fibrations, and the full subcategory spanned by those algebras which present a Markov fibration. Eigil Fjeldgren Rischel2025327Proof Let be an algebra as assumed, and let denote the coequalizer given. By general nonsense there is an induced functor which is moreover an algebra homomorphism---the claim is that this is an isomorphism. By Lemma , pullback to the deterministic part preserves these coequalizers, so this amounts to the claim that the diagram is a coequalizer. But this is true for any algebra of any monad (in fact, the unit gives a splitting of this coequalizer). By general nonsense the fibration associated to an algebra which presents a fibration forms a partial left adjoint to . This left adjoint, by the above, has its image inside and hence there is an adjunction . The preceding furthermore proves that the unit of this adjunction is the identity, which implies that the left adjoint is fully faithful---but by definition it is essentially surjective, finishing the argument. Let us briefly summarize the relationship between Markov prefibrations, Markov fibrations, and stochastic module fibrations at this stage. A stochastic module fibration is a (Grothendieck) fibration over , equipped with some extra structure involving the whole category . Given a deterministic map two objects , and a map , we can think of this as a map parameterized by the fibers . Given a stochastic section , the stochastic module structure picks out a map corresponding to choosing this parameter randomly according to . A Markov prefibration is a category over with a particular unique lifting property. In the above situation, it gives a unique lift of , corresponding to choosing according to and leaving the -coordinate unchangd. By composing this lift with , then with the Cartesian , we get a stochastic module structure on the part of lying over deterministic maps (which is also a Grothendieck fibration). Given a stochastic module structure, there is a way of generating a category over , by freely adding the lifts corresponding to a Markov prefibration, then quotienting by the relations implied by the stochastic module structure. This does not necessarily yield a Markov prefibration. A Markov prefibration is called a Markov fibration if it is presented by its underlying stochastic module in the above sense. Eigil Fjeldgren Rischel 2025 4 8 https://erischel.com/efr-YFV2/ efr-YFV2 /efr-YFV2/ Weak supports Definition Let be a stochastic module fibration. Let be a stochastic section, let be an object, and let be an endomorphism of its pullback. Observe that if there exists so that and factors over , then . We say has weak supports if this implication is an equivalence We will make use of the following lemma: Eigil Fjeldgren Rischel 2025 4 5 https://erischel.com/efr-L7L2/ efr-L7L2 /efr-L7L2/ Lemma Let be a parallel pair in , and suppose both are identity on objects. Suppose moreover this is a reflexive pair, i.e there is some so that . Then the coequalizer in is again identity on objects, and is given on hom-sets simply by the coequalizer of the parallel pair Eigil Fjeldgren Rischel202545Proof The only nontrivial part is to verify that composition is well-defined on the equivalence classes in . It suffices to see that post- and precomposition with a fixed morphism both preserve this equivalence relation. Take some , and . We must show that . But simply write and we are done. Clearly the other side follows by duality, finishing the proof. Eigil Fjeldgren Rischel 2025 4 5 https://erischel.com/efr-TBZZ/ efr-TBZZ /efr-TBZZ/ Construction of Proposition Let be a fibration equipped with a stochastic module structure. Consider the equivalence relation on which identifies two precharts if there exists a map over and a stochastic section of which preserves the section from , so that (note that this makes sense because pullbacks compose). Then: This equivalence relation respects composition, and so defines a category which we denote In , is a coequalizer diagram. In particular, presents a Markov fibration if and only if is a Markov prefibration (in which case is the fibration it presents) If has weak supports, is a prefibration Eigil Fjeldgren Rischel202545Proof Recall that the fibration has fibers whose morphisms are given by tuples (up to a certain equivalence relation). Forming the free Markov prefibration on this fibration, we find that the morphisms are given by diagrams equipped with a map . The two maps carry such a thing to first, the map resulting from forgetting and just composing with to get a span (and composing the sections to get a new section), and secondly, the map with apex obtained by using the stochastic module structure to push down into a map over . It is clear that this is equivalently the equation described in the theorem. This establishes points 1. and 2., since by Lemma we can compute such coequalizers hom-set by hom-set. Now we wish to prove that is a Markov prefibration given weak supports. Since by the coequalizer presentation, its deterministic part is isomorphic to , the fibration property is automatic. It remains to verify that, given a triangle in with the vertical and horizontal maps deterministic, an object and Cartesian maps , there exists a unique lift in . Such a lift is given by a diagram equipped with . By taking the equalizer of the two maps (the section factors over this), we may assume these two are equal, which implies that the pulled-back objects are equal---denote this object . Now the hypothesis is that after postcomposing with the Cartesian map this gives the map . This postcomposition is given simply by postcomposing the leg with the map (and observing that, by functoriality of pullbacks, this does not alter the pulled-back object). Then the claim is that integrating this map down into a map it gives the identity. But by assumption this means we can pull back to some object (lifting the section from ) where the two maps are already equal to the identity. But this pull-back can be applied to the original map as well. But this implies every such map is equal to the one represented by the diagram and the identity on with the map giving the witness, since identities pull back. This map only depends on the underlying hence is indeed a prefibration. It is not apparent whether weak supports are necessary for to be a prefibration. We have not found any counterexample, but in general the equivalence relation on charts is fairly complicated, so it is not apparent how to prove the necessity. We will generally not be too bothered about assuming weak supports instead of the more nebulous assumption that presents a Markov fibration. Eigil Fjeldgren Rischel 2025 9 13 https://erischel.com/efr-26SF/ efr-26SF /efr-26SF/ Lemma Let and be two representatives of charts. Given some possibly stochastic map over and , recall (Lemma ) that we can define , regardless of whether is deterministic or the section of a deterministic map. If there exists such a map , we can always factor it over the pullback as a section followed by a deterministic map. Hence the equivalence relation defining is equivalent to the relation identifying two representatives whenever there exists such an with Eigil Fjeldgren Rischel 2025 4 29 https://erischel.com/efr-XZRN/ efr-XZRN /efr-XZRN/ Remark By construction, for each Markov prefibration there is a canonical functor over , which restricts to an isomorphism on the deterministic part (and in particular preserves Cartesian morphisms). is a Markov fibration if and only if this is an isomorphism. Also by construction, given a morphism of stochastic modules there is an induced functor over . This restricts to on the deterministic part and in particular preserves Cartesian morphisms. Eigil Fjeldgren Rischel 2025 4 27 https://erischel.com/efr-1YS9/ efr-1YS9 /efr-1YS9/ Proposition Let be a stochastic module fibration. Then presents a Markov fibration if and only if does it. Eigil Fjeldgren Rischel2025427Proof Consider a triangle of this for in : where the maps to are deterministic. Suppose given Cartesian lifts of the cospan. These are the same in both cases, coming from Cartesian maps in (which are the same). We must show that there is a unique lift of to a chart in if and only if there is a unique lift to a chart in . Clearly it suffices to prove the "only if" implication, so suppose is a prefibration. By passing to the equalizer as in the proof that is a prefibration, we may assume that any such lift is represented by a diagram where the outer square and the triangle commutes, and is a section. Take such a diagram and let be the map representing a chart. Then the claim is there exists some zig-zag of chart equivalences identifying with . But clearly this is invariant under passing to the fiberwise opposite, and so is also a prefibration. Eigil Fjeldgren Rischel 2025 4 6 https://erischel.com/efr-ANBC/ efr-ANBC /efr-ANBC/ Stochastic charts and lenses Definition We refer to as the category of stochastic charts in . We refer to as stochastic lenses and denote it also Eigil Fjeldgren Rischel 2025 3 28 https://erischel.com/efr-IAFO/ efr-IAFO /efr-IAFO/ Example is a Markov fibration. We have already seen that it is a Markov prefibration, and that the map from the coreflection is full. So it suffices to prove faithfulness. Consider a map in , given by a diagram We can factor the section as , where the first map is just the pairing and the second is a conditional distribution. This induces a factorization of the lift over . By composing the map with this factorization to build the map , we have found a new representative for the same map. Hence every map over has a representative where the residual is . We would like to argue that, since the map is given by the conditional distribution of the composite map it is uniquely determined by it, and thus if two distinct maps in have the same underlying map in they must have equal representatives of this form, and so be identified in the coreflection (which must therefore be isomorphic to ). But of course, the two maps may only be almost certainly equal. In this case, there is a simple fix: instead of taking as the residual, take the subset given by those pairs where has positive probability given . The pairing factors over this, of course, and two maps which give the same distribution really must have the same value on every point. This proves that is faithful and hence an isomorphism. Eigil Fjeldgren Rischel 2025 4 6 https://erischel.com/efr-C5RE/ efr-C5RE /efr-C5RE/ Example , as we have noted, is a Markov prefibration, and hence induces a stochastic module structure on . This structure does not present a Markov fibration. To see this, note that in that case its fiberwise opposite would also present a Markov fibration. Then this fibration, , would be a prefibration whose deterministic part was . But Example shows that this is impossible. Eigil Fjeldgren Rischel 2025 4 6 https://erischel.com/efr-8SXB/ efr-8SXB /efr-8SXB/ Proposition Let be a Markov category with supports. Then the stochastic module induced by presents a Markov fibration. If has conditionals, this Markov fibration is isomorphic to . Eigil Fjeldgren Rischel202546Proof Given a section and , simply take the pullback to the support of . It must be the case that -almost surely, which implies strict equality on the support. Hence by Proposition , presents a Markov fibration. There is an induced map , which we claim is an isomorphism. So consider a map in : This map is in the image of where the map is taken to be a conditional. Note that every map in can be represented in this form, by taking a conditional of given to build a section to . Since conditionals are almost-surely equal, by restricting to the support of , we can find a representative which only depends on the overall map which proves that the map from is faithful, concluding the proof. Continuing from Example , we have the following trivial case: Eigil Fjeldgren Rischel 2025 4 8 https://erischel.com/efr-Z75A/ efr-Z75A /efr-Z75A/ Proposition If is Cartesian (even if it does not admit pullbacks), the definition of Markov fibration over still makes sense, the Markov fibrations are exactly the Grothendieck fibrations, and their fiberwise opposites are simply their fiberwise opposites in the usual sense. Eigil Fjeldgren Rischel202548Proof This is trivial because is simply the identity functor, hence it is monadic (with the identity monad,) hence every fibration/prefibration presents a Markov fibration, namely itself, and is in particular a Markov fibration. The fiberwise opposite is simply given by applying the identity (taking the stochastic module on the deterministic part), taking the fiberwise opposite, then applying the identity again (passing to the presented markov fibration). It is worth noting that, even in the case where is not a prefibration, it may still deserve the name "stochastic lenses". For example the stochastic lenses in can be seen to contain as a full subcategory, even though it does not form a Markov fibration (see Theorem below). Part of the motivation for the theory of dependent optics is to identify a category of stochastic optics which admits all coproducts. If is distributive, satisfies but this coproduct fails to exist in general if the two secondary objects are distinct. The idea is that this coproduct should exist as a family indexed by , where for , and for . Our theory accommodates this example under the mild additional hypothesis of extensiveness Eigil Fjeldgren Rischel 2025 4 6 https://erischel.com/efr-B2P7/ efr-B2P7 /efr-B2P7/ Definition A Markov category is said to be an extensive Markov category if it admits finite coproducts, whose injections are deterministic, and which satisfy the following equivalent conditions: If we let refer to the full subcategory of the slice spanned by the deterministic morphism , we have an equivalence of categories , given by taking coproducts is an extensive category in the usual sense and the inclusion preserves pullbacks along coproduct inclusions. Eigil Fjeldgren Rischel 2025 4 6 https://erischel.com/efr-QAV2/ efr-QAV2 /efr-QAV2/ Proposition Let be an extensive Markov category, let be a fibration which satisfies . Note that this implies admits finite coproducts, and they're given exactly by this pairing. Suppose is equipped with a stochastic module structure. Then preserves the finite coproducts. In particular, has the same coproducts as and preserves them as well. Eigil Fjeldgren Rischel202546Proof This is straightforward to check---the residual splits into by extensivity of , which also implies the section must split as the copairing of a section . By the condition on the fibration, the map in the fiber over splits into a map over and a map over . Using the extensivity again, it is straightforward to see that this decomposition respects the equivalence relation. In particular, both admit coproducts given simply as coproducts in , if is extensive. Eigil Fjeldgren Rischel 2025 3 29 https://erischel.com/efr-XFAO/ efr-XFAO /efr-XFAO/ Proposition Suppose is a Markov fibration so that each pullback functor for can be taken to be bijective on objects, and that these can furthermore be chosen strictly functorial (so that ). Then, writing objects as where is the unique object in which pulls back to under the deletion (note that this means ) we may characterize the fiberwise dual as having hom-sets Eigil Fjeldgren Rischel2025329Proof The correspondence in both directions is obvious by just formally reversing the direction of the map in a representing tuple ---the fact that this assignment respects the equivalence relation follows from the fact that the monad preserves fiberwise opposites. Eigil Fjeldgren Rischel 2025 4 8 https://erischel.com/efr-HWAZ/ efr-HWAZ /efr-HWAZ/ Limits of Markov fibrations The class of fibrations is stable under pullback. This turns into an indexed category, which represents a fibration over ---this is the "global" category of fibrations , whose objects are fibrations and whose morphisms are commutative squares where the top map preserves Cartesian morphisms. By considering universal constructions like limits and colimits in , additional fibrations can be constructed. It would similarly be useful to study limits in the category of Markov fibrations. Moreover, the products in allow one to express notions of internal pseudomonoid---these turn out to be monoidal fibrations, and this is a key part of Moeller and Vasilakopoulou's treatment of the monoidal Grothendieck construction, Reference . Since we want to study monoidal Markov fibrations, we should study their limits. Eigil Fjeldgren Rischel 2025 4 9 https://erischel.com/efr-9VAH/ efr-9VAH /efr-9VAH/ Lemma Let be a Markov prefibration, and let be any functor from another Markov category which preserves deterministic maps. Then the pullback is again a Markov prefibration, and the functor preserves Cartesian maps. Eigil Fjeldgren Rischel202549Proof Since pullbacks compose, is the pullback of along . Since fibrations are stable under pullback, this is a fibration. Now consider a triangle in : with deterministic, and let be Cartesian maps lying over these. We must show there is a unique lift rendering the lifted triangle commutative. By definition, to give such a morphism is to given one over , and the triangle commutes if and only if its image in commutes. The triangle in goes to a triangle of the same class in and the Cartesian maps go to Cartesian maps of the same type. Hence there is a unique lift of of the given type, which is exactly what we needed to show. Eigil Fjeldgren Rischel 2025 4 9 https://erischel.com/efr-Z8X4/ efr-Z8X4 /efr-Z8X4/ Remark Given any functor between Markov categories, we may attempt to define an oplax monoidal structure by pairing the projections. This is not necessarily a natural transformation. However, if it is, it automatically equips with the structure of an oplax monoidal functor. Recall that oplax monoidal functors carry comonoids to comonoids. An oplax monoidal functor between Markov categories preserves the given comonoids if and only if it is induced like this. Hence, there is at most one way to equip a functor between Markov categories with such a structure---it is a property, not extra structure. Call such a functor an oplax Markov functor. Note that oplax Markov functors preserve deterministic morphisms. (Fritz Reference defines a Markov functor to be a strong monoidal functor which preserves the comonoids. Clearly this is a proper subset of our oplax markov functors.) Eigil Fjeldgren Rischel 2025 4 9 https://erischel.com/efr-FNYQ/ efr-FNYQ /efr-FNYQ/ Definition We will denote by the category whose objects are Markov prefibrations , and whose functors are commutative squares where preserves Cartesian maps, and is an oplax Markov functor. We will let denote the category of Markov categories and oplax Markov functors. Note that by Lemma , the forgetful functor is a fibration. There is an obvious functor which carries a prefibration to the pair of its Markov category and its underlying fibration onto the deterministic part. On each fiber, this admits a left adjoint, as constructed in § . By abstract nonsense these left adjoints commute laxly with the pullbacks---that is, given an oplax Markov functor and a map of fibrations over the deterministic part, there is an induced functor over , although this assignment does not preserve Cartesian squares. However this does give a functor , left adjoint to the restriction. The category of algebras over this monad is fibred over with each fiber being the category of stochastic modules over that markov category, and we get a global functor from . In the same way, we get a global functor to which carries each stochastic module to the functor We would like to study the limits in here. At this point, we are forced to consider for a moment a bit of higher category theory. Structures on a category defined "up to isomorphism" generally don't play well together with limits in the category , since they are defined "up to equality". For example, we cannot infer from the fact that have products and the functors preserve them that the equalizer of has products, since given in the equalizer, we have , but not necessarily equality! For the moment we will restrict ourselves to limits of strict (and in particular, strong) monoidal Markov functors, since these always exist. In general one should probably consider some form of homotopy limit, but we will not go into that now. We clearly have: Eigil Fjeldgren Rischel 2025 4 9 https://erischel.com/efr-OMMA/ efr-OMMA /efr-OMMA/ Proposition Let denote the subcategory of strictly monoidal Markov functors (i.e those where the oplaxator is the identity). and admit all products, computed simply as products in . admits all finite limits, and these are preserved by the inclusion into Eigil Fjeldgren Rischel 2025 4 11 https://erischel.com/efr-G0AN/ efr-G0AN /efr-G0AN/ Proposition The category admits all products, and pullbacks along isofibrations. These are simply computed as limits in Eigil Fjeldgren Rischel2025411Proof It is immediately apparent that products (that is, pullbacks over ) of prefibrations are again prefibrations, since the Cartesian lifts can simply be computed coordinatewise, and the uniqueness property checked coordinatewise. Let be a pullback of Markov prefibrations, with an isofibration. (Note that their pullback in is simply their pullback in ). First note that since limits commute, the deterministic part is given by . Thus to prove this is a fibration, it suffices to note that fibrations are stable under pullback along isofibrations. Given and two lifts which are identified in we get two Cartesian lifts . These go to two Cartesian lifts in , and are therefore identified up to isomorphism, but we can lift this isomorphism to and obtain a pair of lifts in the strict pullback---this is a Cartesian lift. Since Cartesian lifts are given by pointwise Cartesian lifts, given a triangle and a stochastic lift we have a unique lift in both . These both go to lifts in ---since such a lift is also unique, they are identified. Hence there is a unique lift in the pullback. Eigil Fjeldgren Rischel 2025 4 10 https://erischel.com/efr-FXBP/ efr-FXBP /efr-FXBP/ Proposition The functor preserves products. For each Markov category with weak conditionals, preserves the terminal object, and pullbacks along isofibrations. Eigil Fjeldgren Rischel2025410Proof The product-preservation is clear from the description of . Let admit weak conditionals. It suffices to show that preserves the terminal object and pullbacks in The terminal fibration is . Clearly the terminal Markov prefibration is so we must show that is an isomorphism. Its morphisms are simply spans with the left leg equipped with a stochastic section , which goes to the composite . As noted before, this is clearly full, by taking , and faithful because the pairing exhibits the equality of this canonical representative with any other. Now let be a cospan of fibrations over , with an isofibration, and consider the pullback . There is a natural transformation which we must show to be an isomorphism. Maps on the left-hand side are given by a span , a stochastic section , and a map in the pullback of the fibers . A map on the right-hand side is given by two spans each equipped with a map, so that they become identified in . Let the apexes of the two spans be . It suffices to consider the case of a span with a common lifting , so that the pullbacks of the two maps to agree. But then the original maps may also be pulled back to have as the underlying span, and thus are in the completion of the pullback. Similarly, given two maps which become identified in the image, we can again use the identifying maps in to identify the original maps, proving faithfulness. This finishes the proof. Eigil Fjeldgren Rischel 2025 4 29 https://erischel.com/efr-TZ9L/ efr-TZ9L /efr-TZ9L/ Proposition For any (pullback-positive) , the monad on preserves products. Eigil Fjeldgren Rischel2025429Proof There is a canonical map . Clearly the deterministic parts are both isomorphic to , and so on this part it is an isomorphism---in particular, bijective on objects. To see it is full, consider an morphism in the codomain, given by a pair of maps , sections and maps in . Then this pair is equivalent to equipped with the pairing and the pullbacks of , which is in the image. Given two maps witnessing equations with another pair of maps, it's easy to see that this lifts to a map witnessing the identity between these, so it's faithful. This concludes the proof. Eigil Fjeldgren Rischel 2025 4 11 https://erischel.com/efr-UF7B/ efr-UF7B /efr-UF7B/ Lemma The property of having weak supports is stable under equalizers in stochastic module fibrations over . If has weak conditionals, it is also stable under finite products (hence all finite limits). Eigil Fjeldgren Rischel2025411Proof (Note that stochastic modules themselves do not admit all finite limits, requiring some sort of isofibration property---we merely claim here that if the limit exists, it again admits weak supports) It is clear that the terminal object has weak supports (regardless of ). Given an equalizer , if is a deterministic map with a stochastic section and is a map in which goes to the identity in , find a factorization so that the image in pulls back to the identity over . Then clearly the same is true for itself. Now consider a product . The point is that given a pair of maps that go to the identity, we can find where the pullbacks are the identity. We form the pullback and use the weak conditionals to find a common lift of the two given sections to this. This gives the required weak supports. Eigil Fjeldgren Rischel 2025 4 26 https://erischel.com/efr-LTEL/ efr-LTEL /efr-LTEL/ Monoidal Markov Fibrations Recall that if is symmetric monoidal, inherits a symmetric monoidal structure. At the same time, if is a monoidal fibration, the fiberwise opposite retains a monoidal structure. Since these monoidal structures play an important role both in compositional game theory (where it would not be much of an exaggeration to say the entire point is to use string diagrammatic syntax to work with games) and in categorical systems theory, it is clearly important to understand the monoidal structure on Markov fibrations. Luckily, as we will see in this section, there are essentially no difficulties in accounting for the monoidal structure. The theory of monoidal fibrations has been developed by Moeller and Vasilakopoulou, Reference , and Shulman Reference . We briefly sketch it here for convenience. There are essentially two available notions of monoidal fibration: For any category , the 2-category admits products, and we can ask for an internal pseudomonoid in this 2-category. This is equivalent to asking for a functor ---in other words, for a monoidal structure on each fiber so that the base-change functors become (strong) monoidal. The global category of fibrations admits products, and we may ask for an internal pseudomonoid here. This is what Shulman calls a monoidal fibration: a fibration where both come equipped with monoidal structures, the fibration is a strict monoidal functor, and Cartesian maps are stable under monoidal product. By a result of Moeller and Vasilakopoulou, these notions coincide in the case where is Cartesian monoidal. Since we are only interested in ordinary fibrations over which is indeed Cartesian, we may apply this result. However, our notion of monoidal Markov fibration will be a modified version of the latter. Eigil Fjeldgren Rischel 2025 4 25 https://erischel.com/efr-CSLQ/ efr-CSLQ /efr-CSLQ/ Monoidal markov prefibration Definition A monoidal Markov prefibration is a markov prefibration equipped with a monoidal category structure on so that is strict monoidal and so that the underlying fibration is a monoidal fibration (i.e so that preserves Cartesian lifts). A braided or symmetric monoidal Markov prefibration is a monoidal prefibration equipped with a braiding or symmetry on so that is moreover a braided monoidal functor. Note that a monoidal Markov prefibration is the same thing as an internal pseudomonoid in the global category of prefibrations. We will not delve further into this point, however. We will need this lemma: Eigil Fjeldgren Rischel 2025 4 26 https://erischel.com/efr-7GH5/ efr-7GH5 /efr-7GH5/ Lemma Let , be a reflexive pair of identity-on-objects, strict monoidal functors. Let be the coequalizer in . Then inherits a monoidal structure making strict monoidal. Moreover, if is symmetric or braided, inherits this structure making a braided functor. Eigil Fjeldgren Rischel2025426Proof The only thing to check is that the equivalence relation on morphisms is stable under tensoring. But this is clear: let . Then witnesses the identification of and . Tensoring on the right is analogous. This finishes the proof for the monoidal structure. In the braided or symmetric case, it is clear that the image of the braiding of in becomes a braiding on it clearly satisfies the coherence equations (being a quotient), and naturality follows by simply choosing representatives and noting that the tensor in is defined by tensoring representatives in . Finally if is symmetric clearly the equation passes to . Eigil Fjeldgren Rischel 2025 4 25 https://erischel.com/efr-85DX/ efr-85DX /efr-85DX/ Theorem The free prefibration monad on has a canonical lifting to Given a monoidal prefibration, its underlying stochastic module acquires the structure of an algebra of this lifted monad. Given an algebra for the lifted monad , acquires a monoidal structure so that is a strict monoidal functor. If moreover has weak supports, this forgetful functor is a monoidal prefibration. Every statement holds also for braided or symmetric fibrations. Eigil Fjeldgren Rischel2025425Proof By Reference , is equivalent to the category of pseudomonoids in . Since the monad preserves products, it must preserve pseudomonoids, which is all we need. Now suppose is a monoidal prefibration. Then its underlying fibration is a monoidal fibration, hence an object of . The claim is that the functor is monoidal. The induced monoidal structure is given on objects by and takes a pair of morphisms in the fiber represented by sections and to . Recalling that the algebra structure is defined by taking to the composite and chasing the below diagram around, it is apparent that the algebra structure preserves the monoidal structure. Now let be a monoidal stochastic module in this sense. It suffices to show that the free prefibration is a monoidal prefibration, by Lemma , and the induced functor will clearly be strict monoidal if is. To construct this monoidal structure on , simply note that since preserves global limits as well, there is an induced monoidal structure on so that the forgetful functor is strict monoidal. Recalling that the Cartesian lifts of to are given by the span and the morphism , it is easy to see by unwinding the definition that these are stable under tensor. If has weak supports, we have already proven that is a strict monoidal functor, and weak supports are equivalent to the claim that it is a prefibration. Since the Cartesian lifts are just the equivalence classes of the Cartesian lifts in the preceding claim that they are stable under tensor implies the same for finishing the proof. The last point is mostly trivial. The product preservation still establishes the lifting to and . Given a braided or symmetric monoidal prefibration, the braidings are Cartesian and in the deterministic part, so the underlying fibration is braided/symmetric and they are preserves by the stochastic module structure. The braiding/symmetry on follows again from Lemma , and there is nothing to show for the last point. Eigil Fjeldgren Rischel 2025 4 26 https://erischel.com/efr-T9F4/ efr-T9F4 /efr-T9F4/ Monoidal Markov fibration Definition A monoidal Markov fibration is a monoidal prefibration which is a Markov fibration. It is braided or symmetric if it is braided or symmetric as a prefibration Eigil Fjeldgren Rischel 2025 4 26 https://erischel.com/efr-KMX6/ efr-KMX6 /efr-KMX6/ Remark Since preserves both global products and fiberwise ones, it induces both a fiberwise monoidal structure and a "global" monoidal structure on . Here we only use the global one. If were Cartesian, the global one would be induced from the local one by, given maps pulling each of them back along the square (and the analogous one for ) and tensoring them over . In a Markov prefibration, of course, these pullbacks are not unique unless is deterministic, but there are "canonical" lifts given by tensoring globally with the (fiberwise) unit map over , and the global tensor is indeed given by the tensor of these canonical lifts (this doesn't provide a noncircular definition of the global tensor, of course). This provides a consistency relation between the two tensor products. Again, we will not dwell on this point. Eigil Fjeldgren Rischel 2025 4 30 https://erischel.com/efr-P4T0/ efr-P4T0 /efr-P4T0/ Markov structure on stochastic charts Proposition Let be a monoidal stochastic module fibration over and suppose each fiber is a Markov category, and this structure is preserved by the pullback functors . Then carries the structure of a Markov category, so that is a Markov functor. If is a Markov prefibration which is also a strict Markov functor, the induced monoidal structure on acquires a fiberwise Markov structure. Eigil Fjeldgren Rischel2025430Proof Every equation in the definition of Markov category involves only deterministic maps, so this can be verified entirely over . Thus this reduces to the claim: given a monoidal fibration over a Cartesian base, if each fiber has a Markov structure, the global monoidal structure has one as well. Given an object a map is by definition a map plus a map . Taking to be the copy map, the codomain there is by definition the monoidal product in the fiber and so we simply use the copying map of the fiberwise monoidal structure. Given a Markov structure on the total category we simply apply this idea in reverse and take the map to be the copy map. The deletion maps can be handled in an analogous way. Eigil Fjeldgren Rischel 2025 4 30 https://erischel.com/efr-FT8J/ efr-FT8J /efr-FT8J/ Corollary Let be a stochastic module fibration, and suppose each fiber has coproducts, and these are preserved by the pullback functors. Then is a Markov category, with monoidal structure given by The Markov structure of Corollary is a generalization of the fact that, if has products and coproducts, and the products distribute over the coproducts, then has products given by . See eg Reference , section 8 for more on this. Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-IMLW/ efr-IMLW /efr-IMLW/ Examples We have already noted several examples throughout. We'll gather a few more here, and also collect a few scattered throughout to make the picture more clear. First, let us make explicit the example of optics, as strongly as it can be stated Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-A6YR/ efr-A6YR /efr-A6YR/ Proposition Let act on . Then has a functor to . The deterministic part admits the structure of a stochastic module fibration. There is an isomorphism . If is itself symmetric monoidal and the action is symmetric (meaning it is given by for some symmetric monoidal functor , see Reference 5.4.3 and 5.5.12), this stochastic module is symmetric monoidal and the isomorphism is an isomorphism of symmetric monoidal categories. Eigil Fjeldgren Rischel2025428Proof We have essentially already seen that the deterministic part is a fibration, with maps over given by , and with the pullback functors acting by reparametrization. Since a map with deterministic marginal on is always equal to the independent pairing of we can slide the former through and identify each optic over a given with a map and note that this map is conversely an invariant of an optic, since it is obtained by postcomposing with . Given , objects and a map over classified by a section act by reparametrization. It is clear that this gives the structure of a stochastic module. The functor from optics takes to the span equipped with the obvious map that simply forgets . Note that every chart is equivalent to one of this form (given and the map exhibits the required equivalence), hence the functor is full. Moreover, note that each chart is associated with a well-defined optic, given by the maps . Is is straightforward to see both of these maps are preserved by chart equivalence. This gives an inverse to the functor, proving it is faithful. Since it is identity on objects, this finishes the argument. In the symmetric monoidal case, it is immediately clear that the fibration on is symmetric monoidal. Since the action is symmetric monoidal, given maps and it is clear that composing to get maps then tensoring and composing with the diagonal to get gives the same map as tensoring, then using the map . Hence we have a symmetric monoidal module. It's straightforward to see the functor is symmetric monoidal, and that finishes the argument. Slightly orthogonally, we have the following comparison between and : Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-K6NM/ efr-K6NM /efr-K6NM/ Theorem Let be any pullback-positive Markov category. Then is a Markov prefibration which thus induces a stochastic module structure on . Writing simply for , we have: There is a functor which is fully faithful. Dually there is a functor which is fully faithful. and both admit symmetric monoidal structures, which make the functors strict symmetric monoidal, as well as the functors strong symmetric monoidal. If is extensive, this functor preserves the coproducts and both admit all finite coproducts. If moreover has conditionals and supports, Thus we have our previous claim that contains . We also have some examples of an analytic flavor: Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-TO9K/ efr-TO9K /efr-TO9K/ Example Given a compact Hausdorff space , a Banach space bundle is a space over , , equipped with a fiberwise (complex) vector space structure , , so that there exists a cover of so that for each , there exists a Banach space and a homeomorphism over which is moreover linear in each fiber, where is equipped with the norm topology. Note that this determines a local norm on each (and in particular each ) up to equivalence (but no stricter than that). In particular each is a Banach space. A morphism of Banach space bundles is a continuous map over which is linear on each bundle. Note that this implies that on a suitable cover the maps obey for some , for any local norms inducing the topologies, and hence by compactness there exists some so that for each . If is a continuous map, there is a pullback functor . The Grothendieck construction of this gives a fibration Note that Tychonoff spaces include all compact Hausdorff spaces. Therfore consider the full subcategory spanned by these. The fibration admits the structure of a stochastic module: given , a kernel, and a linear continuous map given there is an induced function given by . Since this is bounded (being continuous on a compact space) it is (Bochner) integrable, define to be this integral. This example is analogous to optics for the action of categories of markov kernels on categories of vector spaces (as in Proposition ). Note that we do not expect this type of example to present a Markov fibration. The reason is simply that, given a parametrized linear map and a measure on , the fact that the expectation map is the identity by no means implies that the original map is almost surely the identity or anything like that. If and has positive probability, this can be canceled out by . This is impossible for probability kernels. If we add an assumption of positivity, it seems plausible that examples of this type will present Markov fibrations---but of course, that brings us quite close to categories of Markov kernels in any case. Note that, as in this example, we do not generally expect to yield a Markov fibration if is not another Markov category (and not even then in general, as the case of shows).---for these, we expect to need a sort of positivity in the fiber as well, which restricts us to things that look like probability kernels. Eigil Fjeldgren Rischel 2025 5 2 https://erischel.com/efr-O6GQ/ efr-O6GQ /efr-O6GQ/ Stochastic module of -algebras Proposition Let be a representable, positive Markov category so that admits intersections and the probability monad preserves them. Then each slice inherits a monad structure given by . This is pseudofunctorial in . Moreover, the stochastic module structure on extends to a stochastic module structure on the fibration representing the pseudofunctor . Eigil Fjeldgren Rischel202552Proof The monad is induced by the adjunction . Let (deterministic). For abstract reasons there is a natural transformation . Writing this out, we find By representability, the unit is a monomorphism. Hence a map into is precisely a map in into so that the marginal on is deterministic. But by pullback-positivity this is precisely a map (in ) into . Let , two -algebras , and a map which is a homomorphism for the induced -algebras, and a map be given. Note that by the above, . Then the induced map is given by We must show this is a -homomorphism. Let us simplify by working internally to ---thus we have a Cartesian category equipped with a strong commutative monad , a map , and a map which is a parametrized algebra homomorphism, in the sense that the diagram commutes. Now we must show the map is a -homomorphism. Write for the structure maps of the two algebras. Consider this diagram: The triangle at the top left commutes because is strong. The square to the right does not commute in general---however, since is commutative, the composite maps agree. Since the map factors over this, we may replace one edge of this square with another. The square to the right of that is simply applied to the previous diagram, and so commutes by assumption. The "triangle" under that is just two copies of the same maps, so commutes. The square on the left of the diagram commutes by functoriality of product. The square to the right of that commutes again because is strong. Hence the outer square commutes, which is precisely the homomorphism property we wanted. It is apparent that, if are free algebras, this restricts to the stochastic module structure of (viewing as the Kleisli category of ). But since every algebra is a coequalizer of free algebras, it follows that the action on general algebras is determined uniquely by this. This implies the equations of a stochastic module. As in Example , this cannot be expected to come from a Markov prefibration in general. The vast majority of examples seem to occur as subcategories of stochastic modules of the form given by Proposition (of course, is just the subcategory spanned fiberwise by the free algebras). In fact, since a stochastic module necessitates in some sense an action of on the objects of the fiber, it seems they do all have this form in a generalized way, although we have not found a better way to make this precise than the existing definition of stochastic module. Eigil Fjeldgren Rischel 2024 6 13 https://erischel.com/efr-000D/ efr-000D /efr-000D/ The construction in generic 2-categories Eigil Fjeldgren Rischel 2024 7 22 https://erischel.com/efr-003P/ efr-003P /efr-003P/ Introduction Myers' theory of categorical systems theory (see § ) gives a rich categorical structure to a wide variety of types of dynamical system. The central idea can be summarized by saying that there are two different, but tightly related, notions of morphism at play in dynamical systems. Open dynamical systems themselves involve a bidirectional information flow, captured by the notion of lens and composition of such lenses describes the composition of subsystems into systems. But morphisms between systems are unidirectional, captured by the notion of chart . Algebraically, the relationship between these two notions is that they assemble into a double category, which indexes the category of systems. There is another double category involving lenses which has been considered in the categorical study of systems. That is the double category of parametrized morphisms, applied to the monoidal category of lenses. These have been studied as an abstraction for gradient descent in several papers, see eg Reference , Reference , (the idea goes back to Reference ). Given any action of a monoidal category on another category (most simply, if is monoidal it acts on itself,) we obtain a category of parametrized morphisms (where ,) and these turn out to be extremely useful. We will mention two applications: A parametrized lens is essentially what is called a learner in Reference ---it contains the information necessary to compute a new parameter value given an existing and a sample pair , as well as the additional information required to compose such things. Thus the functoriality of backpropagation can be derived from two facts: the reverse derivative defines a monoidal functor and the construction taking a category to its category of parametrized maps, is itself functorial. This viewpoint has been significantly developed in Reference and other papers. An open game, in the sense of Hedges Reference , is almost the same thing as a parametrized lens , equipped with a subset . Given a context for the game---that is, a state (describing the state of information when the decision is made) and a continuation (describing how decisions map to outcomes ,) we obtain a function , and we say is an equilibrium strategy if . Since and this neatly captures the extra data of an open game in terms of the category of lenses. The potential of this idea as a generalized approach to "cybernetic systems" is explored in Reference . In this chapter, we will develop the theory of the category of parametrized maps. We will begin by reviewing the existing literature briefly. We will describe a double categorical version of this category---this does not seem to have appeared in the literature yet, although it has been folklore for at least a few years (and there is nothing complicated about this construction, certainly). The remainder of this chapter will be dedicated to lifting this construction to a generic 2-category (with the above being the specialization to ). We will derive this lifting using the machinery of 2-category theory. We will see how this generalization accounts for much structure which can be seen to exist on , such as its symmetric monoidal structure (assuming are symmetric monoidal). But the true application of this will be in § , where we use this to construct a triple category of open dynamical systems. The definition of the double category involves two types of categorical structure with which the reader may be unfamiliar---actegories, which are the input to the construction, and pseudo double categories, which are the output. Since we will shortly introduce the abstract internal versions of these, internal pseudomonoid actions and internal pseudocategories, we will not give a separate introduction here. The reader who is unfamiliar with these should refer to Reference for actegories, and Reference for (pseudo) double categories. A reader who simply needs a definition may look at Definition and Example . Eigil Fjeldgren Rischel 2024 7 2 https://erischel.com/efr-002H/ efr-002H /efr-002H/ The Para Construction as a double category In many different situations, we want to understand some morphism as parametrized by some data. For example, in machine learning one tries to find a function with some desirable behaviour by choosing a parametrization and searching for some so that has this behaviour (for example by gradient descent on ). In situations where we want to understand as being built up as a composite of multiple functions (for example, the layers of a neural network), it is convenient to introduce a category of parametrized morphisms, where the composition combines the parameter spaces of each composite map. We can do this in a general setting with the following definition: Eigil Fjeldgren Rischel 2024 7 2 https://erischel.com/efr-002G/ efr-002G /efr-002G/ Definition Let be a monoidal category, and let be an action of on another category . Then the construction is the category where Objects are objects of Morphisms are tuples up to the natural notion of isomorphism. Composition is by tensoring the parameter objects and composing. The fact that the parametrization object may live in a different category than the domain and codomain objects, which initially seems like a superfluous generalization, is in fact highly useful. For example, we will often want to consider parametrized morphisms where the parameter space is decorated with some additional data, for example a probability distribution. In most such cases, the category of spaces decorated with such data will act on the category of spaces without such data (simply by forgetting the data and tensoring), and hence we can realize such parametrized morphisms as examples of this construction. We may also want only Euclidean parameter space (so that we can run gradient descent simply), but allow the parametrized morphism to go between more general manifolds. , and generalizations of it, have been introduced many times. One of the earliest occurrences is by Hermida and Tennent, Reference , in the special case of a symmetric monoidal category acting on another such via a functor (by a result of Capucci and Gavranović, Reference , every "structure-preserving" action of a symmetric monoidal category on another has this form). Hermida and Tennent actually give a very intuitive universal property of : it is freely generated by adding a morphism for each , subject to the equations that this must be a monoidal natural transformation. This type of idea actually goes all the way back to Pavlović in Reference . The notation was introduced inReference , where the special case of parametrized morphisms of euclidean spaces was used to study gradient descent. In Reference , a bicategorical variant (replacing the quotient by isomorphism in the above variant in the obvious way) is introduced. Eigil Fjeldgren Rischel 2025 4 29 https://erischel.com/efr-MN7C/ efr-MN7C /efr-MN7C/ Example Let be a monoidal category, and let be any functor. Recall that is the category whose objects are pairs and whose morphisms are so that . If is lax monoidal (for ,) acquires a lax monoidal structure making the forgetful functor strong (even strict) monoidal. With the action induced by this functor, the bicategory has morphisms given by maps and maps given by reparametrizations which preserve the decoration . For example, let be a Markov category and let . Then morphisms are parametrized maps equipped with a measure on the parameter space. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-WEST/ efr-WEST /efr-WEST/ Example If regarded as a discrete 2-category, as noted, a pseudomonoid action is just a monoid action in the ordinary sense---that is, a monoid , a set and a function so that . The para construction is then the action category, whose objects are the points of and whose morphisms are elements so that (with multiplication as composition). Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-QH7O/ efr-QH7O /efr-QH7O/ Example Let be the 2-category of -enriched categories, functors and natural transformations. Recall that a ring is the same as a one-object category enriched in , and it admits a monoidal structure if and only if it is commutative (this is not particular to -enriched categories). An enriched -action on an -category is then an -module structure on each hom-set so that composition is -linear in each variable. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-KL7V/ efr-KL7V /efr-KL7V/ If a category has coproducts, then acts on it via . (This is the -enriched case of what in enriched categories is called a copower or tensor, the dual of the power objects from Example ). The morphisms of are pairs , which compose in the obvious way. Note that this definition clearly makes sense even if does not actually have coproducts. This is an example of another construction which has been called , which takes a monoidal category and a -enriched category and constructs a double category where the morphisms are pairs . It is not hard to see that this also extends to a double category in the same way, but we do not presently know the correct definition of internal enriched object that would replace pseudomonoid actions to replicate our general theory for this case. (note that the literature contains a notion of internal enriched category, Reference , but these are categories enriched over internal monoidal categories---that is, the ambient category is a 1-category and the categorical structure of the base of enrichment is formulated on top of this, not as part of the structure of the objects of the category ) There is a very natural double categorical version of , where the vertical morphisms are just the unparametrized morphisms of , and the 2-cells with this boundary: =4pt, Rightarrow, from=0, to=1, "g"] \end {tikzcd} ]]> are morphisms (if is parametrized by , by ) so that the square commutes. This notion is not original---we learned of it from Matteo Capucci---but it seems to have remained somewhat on the level of folklore. Our main goal for this section will be to construct this (pseudo) double category, and show that it is functorial---that is, given a homomorphism of actions, there is an induced functor between double categories. In fact, we will show that this construction works in any 2-category. Later we will apply it to a pseudomonoid action in (strict) double categories to construct a pseudocategory internal to double categories---our triple category of bimachines. Eigil Fjeldgren Rischel 2025 4 29 https://erischel.com/efr-55CC/ efr-55CC /efr-55CC/ Theorem Let be a monoidal category, and let be a category with an -action. Then there is a pseudo double category with , with domain and codomain given by the two projections to The identities map is given by where is the left unitor of . The horizontal composition map is the composition in given their composite is given by The horizontal composition of 2-cells is defined analogously. Moreover, if is strict homomorphism of -modules, there is an induced pseudofunctor . If is a strict monoidal functor, then regarding as an -module along this map, there is an induced pseudofunctor . These combine into a 2-functor between the 2-category of actions and strictly linear functors and the category of pseudo double categories and strict double functors. This functor preserves (strict) limits. Eigil Fjeldgren Rischel2025429Proof Note that every actegory is equivalent to a strict action of a strict monoidal category (in the sense that there exists strong monoidal equivalence and an equivalence such that these maps together form a map of actegories). It follows that it suffices to show unitality and associativity for our composition in the strict case (since these are plainly preserved by equivalence). Let , be a strict monoidal functor, and let be a linear functor "over ", that is a (strict) -linear functor when is viewed as a -actegory along . Then the induced functor is given by on the vertical category, and on the horizontal category by where the isomorphism is the linearity---noting that the -action on is precisely acting by . By naturality of the lineator it is straightforward to see that this is a functor, and it clearly preserves domain and codomain. It preserves units (strictly!) by the compatibility between the unitor and lineator. It remains to see that this preserves composition. For brevity, denote the induced functor from here. It suffices to verify this for strict actions. So the composite of is given by the composite Given two such composable maps, call them , the two objects are given by ,where are respectively the maps across the top and bottom of this diagram: This proves that preserves the composition strictly, as desired. (The rightmost triangle commutes by definition, and the leftmost rectangle is a lineator coherence) Since the strict limits in and are computed levelwise, it suffices to observe that the slice category also preserves limits, being a weighted limit itself. Eigil Fjeldgren Rischel 2025 4 29 https://erischel.com/efr-HUEE/ efr-HUEE /efr-HUEE/ Remark Note that comes equipped with a strict functor to (viewed as a double category). In fact, one can recover the action from this data: the object is characterized up to isomorphism by its universal property: parametrized maps with parameter are in bijection with maps parametrized by . Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-9I1M/ efr-9I1M /efr-9I1M/ 2-Limit sketches Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-L9JB/ efr-L9JB /efr-L9JB/ Weighted limit Definition Let be a 2-category, let be a 2-diagram in it, and let be another 2-functor. A limit of weighted by {W} is an object equipped with a natural isomorphism of categories (natural in ). Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-K3YI/ efr-K3YI /efr-K3YI/ Particular weighted limits Example If is constant at the point, and is an 1-category, then a weighted limit is simply a in the underlying category so that there is additionally an isomorphism of categories . Note that this is stronger than just a limit in . We will refer to limits of this form by their ordinary names---speaking for example of pullbacks, products, and so on. If , , and the universal property of the weighted limit is that . In this case we write for this limit if it exists, and call it a power of by . If and the weighting is given by (with the obvious inclusions), then the weighted limit is called the comma object and will be denoted . Note that if this is precisely the ordinary comma category. The analogue for a cone on an object in the setting of weighted limits is called a cylinder: it is a natural transformation . A cylinder on induces a natural transformation and we say it's a limit cylinder if this is an isomorphism. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-40Q5/ efr-40Q5 /efr-40Q5/ 2-limit sketch Definition A 2-limit sketch is a small 2-category equipped with a (small) set of cylinders . A model of the sketch in is a 2-functor which carries each cylinder in to a limit cylinder. We write the category of models and natural transformations (leaving the set of cylinders implicit). If we write simply . Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-D7EH/ efr-D7EH /efr-D7EH/ Lemma Let be a 2-limit sketch. Then the functor given by is fully faithful, and its essential image consists of those functors so that each is representable, Eigil Fjeldgren Rischel202554Proof First note that the codomain can be identified with the subcategory of spanned by those where each is a model. Since the Yoneda embedding preserves limits, the functor from clearly lands inside here. Since is itself a full subcategory of the functor category this is fully faithful. Clearly for each model and for each the functor is representable, by . Conversely, if is such that each is representable, then the currying of factors over , and since the Yoneda embedding preserves those limits that exist, this factorization must be a model as well. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-9O7K/ efr-9O7K /efr-9O7K/ Proposition Let be limit sketches and suppose given a functor which preserves limits and is accessible, that is it preserved -filtered colimits for some . Then admits a left adjoint . In particular, Eigil Fjeldgren Rischel 2025 3 21 https://erischel.com/efr-ZRUW/ efr-ZRUW /efr-ZRUW/ Pseudocategories Given a 2-category , there is a natural loosening of the notion of "internal category in ", given by requiring that associativity and unitality hold only up to chosen coherent isomorphism 2-cells. This is the notion of pseudocategory. We refer to Reference for a thorough description of this concept (as well as a review of the earlier literature). However, we will review the basic facts here for convenience. Eigil Fjeldgren Rischel 2025 3 21 https://erischel.com/efr-ZRUX/ efr-ZRUX /efr-ZRUX/ Internal pseudocategory Definition Let be a 2-category with pullbacks. Then an internal pseudocategory in (or simply a pseudocategory in ) consists of the following data: Two objects Morphisms so that . A morphism , so that Isomorphism 2-cells , , and Satisfying the following equations: , And so that the following diagrams commute: A homomorphism or strict functor of pseudocategories is a pair of morphisms which commute with all the structure---that is, , , and so on. There is a clear notion of natural transformation of homomorphisms. We write for the 2-category of pseudocategories, homomorphisms and natural transformations in . It is important to note that, although this is a weakening of the definition of internal category, pseudocategories are themselves a strict concept---they are defined in terms of equations that must hold up to strict equality. Thus for example there is an enriched monad (a 2-monad) on whose strict algebras are the pseudo double categories. Eigil Fjeldgren Rischel 2025 3 21 https://erischel.com/efr-ZRUY/ efr-ZRUY /efr-ZRUY/ Example An internal pseudocategory in is, as mentioned above, a pseudo double category. An internal pseudocategory with terminal is the same thing as an internal pseudomonoid. If we fix a specific terminal object and require , this is an isomorphism of 2-categories (both for the categories of pseudomorphisms and homomorphisms). An internal pseudocategory with identities is the same thing as a (strict) internal category. In particular, internal pseudocategories in discrete 2-categories are merely internal categories. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-LTJQ/ efr-LTJQ /efr-LTJQ/ Example Let be the 2-category of monoidal categories, strict monoidal functors and monoidal natural transformations. Note that this has pullbacks. Then the objects of are a stricter version of monoidal pseudo double categories: they are pseudo double categories where both carry a monoidal structure so that are strict monoidal functors and are monoidal natural transformations. Here we are already beginning to feel the limitations of this internalization a bit---really we should ask for , at least, to be only a strong monoidal functor. But we move on for now with this problematic definition. Of course, there is a very crucial notion of pseudofunctor between pseudo double categories. This has an internal formulation in terms of pseudomorphisms: Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-8YZ9/ efr-8YZ9 /efr-8YZ9/ Pseudomorphism of pseudocategories Definition Let be internal pseudocategories in a 2-category . A pseudomorphism consists of the following data: Morphisms , in Isomorphism 2-cells and So that the following equations hold: And so that the following diagrams commute: One should not spend too much time looking at Definition . This is clearly a horrible concept, and we will prefer to work around it rather than referring to it explicitly. It may be a useful exercise to convince oneself that, when and the pseudocategory is a pseudomonoid---eg a monoidal category---this coincides with the notion of strong monoidal functor. One can also define lax and oplax functors between pseudo double categories, but we will not need that concept here. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-7ZZG/ efr-7ZZG /efr-7ZZG/ Proposition There exists a -limit sketch whose strict models are internal pseudocategories, and whose strict natural transformations are homomorphisms. Eigil Fjeldgren Rischel202554Proof There is nothing surprising about this to those familiar with limit sketch presentations of other algebraic gadgets, but for completeness we give an explicit construction here. Take to have objects . The 2-category is freely generated by these objects and the following data: Morphisms Morphisms for so that commutes. Morphisms and satisfying Further morphisms, and invertible 2-cells as in the following diagrams: =4pt, Rightarrow, from=1-2, to=2-1] \arrow ["m", from=1-2, to=2-2] \arrow ["m"', from=2-1, to=2-2] \end {tikzcd} ]]> =3pt, Rightarrow, from=0, to=1] \arrow ["\rho "', shorten <=3pt, shorten >=3pt, Rightarrow, from=2, to=1] \end {tikzcd} ]]> and all the other induced morphisms appearing in Definition , subject to those equations holding. Equations asserting that the morphisms and 2-cells with codomain 1]]> behave as expected under postcomposition with the pullback projections. The prescribed pullback cones are the diagrams involving for . An explicit description of this sketch seems not to have appeared in the literature until Reference (example 5.13). Note that they construct a richer object, what they term an -sketch, which includes the information that the domain and codomain are "tight" and must be preserved strictly by pseudofunctors, but composition and identities are "loose" and may be preserved only up to natural isomorphism. This idea will be highly useful later, although we will not go into a full treatment of their theory. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-NI76/ efr-NI76 /efr-NI76/ Actegories Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-OFNV/ efr-OFNV /efr-OFNV/ Actegory Definition Let be a monoidal category. A -actegory (also -module, -action) is a category equipped with a functor called the action (written infix, ), and natural isomorphisms satisfying the following coherence equations: Actegories are a very natural concept, and as one would expect their history goes back a long way. The concept has been considered by Benabou all the way back in Reference , and used many times since them. We will not delve too deeply into their theory here---see Reference for a thorough treatment. Just as pseudo double categories, actegories really make sense in every 2-category, as a weakening of monoid actions. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-IOLJ/ efr-IOLJ /efr-IOLJ/ Pseudomonoid action Definition Let be a 2-category. A pseudomonoid action internal to consists of the following data: An internal pseudomonoid , an object , a morphism natural isomorphisms and , satisfying the coherence equations from Definition . A strict homomorphism of actions is a pair so that is a strictly monoidal functor and preserves the action strictly, i.e , etc. Note that this makes sense even if the monoidal category or action is not itself strict. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-6S13/ efr-6S13 /efr-6S13/ Proposition There is a limit sketch whose models are tuples of a pseudomonoid , an object and an action of on . The strict natural transformations between models are strictly linear functors. Eigil Fjeldgren Rischel202554Proof As above, there is nothing difficult about this. Let contain as a subcategory the sketch of pseudomonoids , along with three additional objects projections , and limit cones expressing these as the products and respectively, a morphism and isomorphism 2-cells and , subject to the coherence equations in Definition . Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-I33G/ efr-I33G /efr-I33G/ Discrete pseudomonoid Example If is an ordinary category with finite products (viewed as a discrete 2-category), a pseudomonoid is simply an internal monoid, and an action is just a monoid action in the ordinary sense. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-ARIX/ efr-ARIX /efr-ARIX/ Monoidal and symmetric monoidal actegories Example In Reference , the authors study actegories with extra monoidal structure. Although they do not introduce pseudomonoid actions in a general 2-category, they study actegories internal to monoidal categories, which they call monoidal actegories. It is straightforward to check that pseudomonoid actions in agree with their notion: such an action consists of a braided monoidal category , an ordinary monoidal category , a monoidal functor and monoidal natural transformations satisfying the coherence equations for an actegory. (Note that since the functor appears on one side of , to speak of being a monoidal natural transformation we must have a monoidal structure on ---this makes into a braided monoidal category). For the case of symmetric monoidal actegories, since is cocartesian, . Given two symmetric monoidal categories Reference show that an action is simply given by a strong symmetric monoidal functor (and in this case the action is ). Unwinding this construction we see that is bi-equivalent to a category which has as objects strong (but not strict) symmetric monoidal functors and morphisms squares of symmetric monoidal functors, where the top map is strict, and which commute strictly. Again, as expected, the notion of strict linear morphism is usually too strict. One can expect an equation like to hold only up to coherent isomorphism. Let us now make this clear: Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-L5QG/ efr-L5QG /efr-L5QG/ Pseudolinear map Definition Let be a 2-category, and let be internal pseudomonoid actions. A pseudolinear map (or just linear) is a pair of maps , equipped with a pseudomonoid pseudohomomorphism structure on and a natural isomorphism satisfying the following coherence conditions (note that writing functors using element-notation like this is well-defined) In Reference , these are defined in two steps: first linear functors for the same (those with ) are considered. Then, given a strong monoidal a functorial assignment of an -action to every -action is constructed, and the full category of actions is defined as the Grothendieck construction of this. There is nothing preventing this from working in the setting of a general 2-category, and it is straightforward to verify that our notion of linear morphism of actions agrees with theirs. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-GZH4/ efr-GZH4 /efr-GZH4/ for a general 2-category We are now almost ready to prove: Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-I897/ efr-I897 /efr-I897/ Theorem Let be a 2-category which admits pullbacks, products and comma objects. Then there is a functor from the 2-category of pseudomonoid actions and pseudolinear maps to the 2-category of pseudocategories and pseudofunctors, with and . This functor preserves strict maps, limits, and filtered colimits. The main ingredient missing is a characterization of the respective notions of pseudomorphism in terms of the limit sketches. This we do now: Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-NZ0X/ efr-NZ0X /efr-NZ0X/ Proposition Let be a 2-category and let be pseudocategories, represented as models of the theory. Then a pseudonatural transformation which is strict on and the limit projections is equivalently a pseudomorphism between the pseudocategories. Now let be pseudomonoid actions. Then a pseudonatural transformation which preserves the product projections strictly is equivalently a pair of a pseudomorphism and a functor which is pseudolinear with respect to the action of on . Eigil Fjeldgren Rischel202554Proof Both of these are essentially a matter of unwinding the definitions. Let us start with pseudocategories. The requirement that is strict on the product projections amounts to requiring that is given strictly as the pullback and not merely up to 2-isomorphism. With this, is just a naturality transformation for , and is a naturality transformation for . Note that since every map in is induced from and the pullback structure, this means the other naturality transformations are determined by The composite is the naturality transformation for the map , and analogously for the composite . Hence the hexagon is a pseudonaturality coherence for the 2-cell . Analogously the two squares can be obtained as coherence 2-cells. Conversely, these suffice to make a pseudonatural transformation, again since all the other 2-cells are generated by . Now let us look at pseudomonoid actions. As above, such a pseudonatural transformation is determined by the functors and plus some 2-cells involving these and their products. As a special case of the above when we find that the restriction of the pseudonatural transformation to the pseudomonoid part is equivalently a strong monoidal functor. Now the required lineator is a naturality square for the multiplication map . As above this determines all the other naturality transformations because all the other maps are generated from (and the pseudomonoid structure) using the product structure. Also analogously to the above argument, the coherence pentagon is a pseudonaturality coherence for the natural isomorphism , and the coherence triangle for . Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-T667/ efr-T667 /efr-T667/ Remark This characterization of the notions of pseudomorphism cannot help but seem a little ad hoc. The basic idea here seems to go back to Power in Reference . After constructing an enriched Lawvere theory (which is just a special kind of limit sketch) corresponding to any finitary enriched monad, he notes that in the -enriched case the pseudomorphisms of monad algebras correspond to exactly those pseudonatural transformations which preserve the products strictly. An elegant theory which generalizes this basic idea has recently been developed by Bourke, Ko, and Varkor in Reference (as we mentioned briefly before). Their theory involves decorating the sketch with a set of tight morphisms, which the transformations must be strictly natural with respect to. Beyond developing a general theory of this (not specialized to a few examples as above,) they also show how to construct the categories of (op)lax homomorphisms, and develop a commutativity of internalization principle for their models. This could be profitably applied in our case to understand better, for example, the implied notion of monoidal triple category, but we will not pursue this here. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-0BHI/ efr-0BHI /efr-0BHI/ Proof of Theorem Proof First recall that can be identified with the subcategory of spanned by the levelwise representable presheaves. Note that for it suffices to verify representability for since the rest are pullbacks of these, and pullbacks of representable presheaves are again representable since admits pullbacks. Postcomposition with the previously-constructed functor gives a functor . Clearly this functor preserves representability (again, since limits of representable functors are representable). This gives the desired functor . Under the identification of with a subcategory of it is clear that the pseudonatural transformations of models correspond to those pseudonatural transformations which are strict on morphisms in . This implies the full category of models and pseudonatural transformations can be identified with the subcategory of spanned by those 2-functors which come from models---that is, must factor over . (To be clear: the morphisms in are strictly natural transformations between functors , but each component of such a natural transformation is a pseudonatural transformation of models). Under this identification, it is clear that pseudofunctors correspond to those natural transformations in which are valued in pseudofunctors, and analogously for pseudolinear morphisms and maps . Hence it suffices to observe that the -valued version carries pseudolinear maps to pseudofunctors. Note that is given objectwise as a finite limit. Moreover, since the limit sketch of pseudocategories only involved finite limits, the class of pseudocategories is stable under filtered colimits in . Hence commutes with filtered colimits, and is in particular accessible. Hence it admits a left adjoint where is the restriction of the left adjoint of . Both forming the hom-category and precomposing with are 2-functors, and as such preserve pseudonatural transformations. Thus it only remains to observe that also preserves the partial strictness property, but this clear. We have not yet given any thought to functoriality in , but passing through the constructions, it is apparent that we have: Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-TY4G/ efr-TY4G /efr-TY4G/ Proposition If is a 2-functor that preserves finite 2-limits, the square commutes up to strict natural isomorphism. (In particular, the square involving the subcategories of strict homomorphisms also commutes up to natural isomorphism). (It may seem wrong that, after working strictly all this time, this square commutes only up to natural isomorphism, but in fact that is the strict notion---the weak version of this statement would be that it commuted up to natural equivalence. Note that eg. the comma objects are only characterized up to isomorphism, so this is really the best we can hope for) The main problem with Theorem is that for many 2-categorical notions of interest, requiring (for example) the action to be a strict homomorphism of whatever structure under consideration is too strict to work, while working with the full category of pseudomorphisms prevents the pullbacks required for pseudocategories from existing. Thus for example, an internal pseudomonoid in is a commutative monoidal category, which is far too strict for most purposes---generally speaking, symmetric monoidal categories can not be strictified into commutative ones. In § , we will want to construct a symmetric monoidal "triple category"---that is, a symmetric monoidal pseudocategory in strict double categories. We will manage this via the preceding by noting that since preserves products, it carries (symmetric) pseudomonoids to pseudomonoids---in other words, we can apply the internalization the other way around, taking pseudomonoids in actions rather than actions in pseudomonoids. This works because strict products still exist in the category of pseudohomomorphisms, but for a more general categorical structure, we would be in trouble. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-GFG0/ efr-GFG0 /efr-GFG0/ Open Games with external choice in Markov Fibrations Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-7EHV/ efr-7EHV /efr-7EHV/ Introduction Game theory is a field of economics which studies mathematical models of human decisionmaking. Classically, game theory is particularly interested in the behavior resulting from individual agents optimizing simple objectives (such as the expected value of some real-valued function of each players' decision) when multiple such players interact. Although the analysis of games, obviously, has a long prehistory, the field was put on its modern theoretical footing by Von Neumann in 1928 (Reference , see also Reference for a more thorough treatment from this era). The basic problem studied here is this: given two finite sets of strategies---that is, the choices available to the two players---and a function which assigns to a pair of such choices a score, which player 1 seeks to maximize and player 2 seeks to minimize, what can be said about the rational decisionmaking of each player? Suppose each player has access to some source of randomness, so that they may each choose a distribution on their respective strategy set to play. This is called a mixed strategy. Suppose player 2 has the opportunity to move knowing the distribution chosen by player 1 (but not the value drawn from it), and suppose he prefers to minimize the expected outcome of the game. Writing for for brevity, if player 1 selects the distribution , clearly player 2 must select the distribution . Knowing this, player 1 will select the distribution and the expected score of the game will be If the selection happens in the other order, of course, the result will be Clearly choosing knowing your opponent's (mixed) strategy can not be a disadvantage compared to choosing with no information, and so we have Von Neumann's great minimax theorem is that these values agree, and this implies that by choosing as if your opponent would know your (mixed) strategy, you obtain a result as good as you could have obtained if you knew your opponents' strategy---hence neither player could improve their outcome, even if they had the advantage of greater information. This is a strong argument for the optimality of such decisions. The key assumption is that the players are perfectly opposed, that is, player 2 seeks to minimize precisely the value that player 1 seeks to maximize. Nash in Reference extended the theory to the more general class of games where players simply each have their own utility function, although it should be noted that Nash merely proved the existence of equilibria---that is, strategy sets where no player can improve their situation by switching strategies. It is computationally intractable (Reference ) to identify Nash equilibria in an arbitrary game, and so to apply the theory we're forced to analyze each game in an ad hoc way. We will not go intro a serious review of game theory here, we mention the above details mainly to contextualize our future definitions. For a modern reference on game theory see eg Reference , or Reference for a textbook treatment. In his thesis (Reference ), Hedges introduced a novel approach to game theory which he named compositional game theory, studying objects called open games. The idea of open games is to describe a type of "partial" game which has an interface to the world---some part of the payoff function being dependent on an undetermined environment, into which hole can be inserted another game. This is the sense in which they are open. Based on these, one can build a complex game up out of simpler subparts, as well as leverage a string diagramattical syntax to analyze games. The original definition of open game cannot help but seem somewhat ad hoc. It was quickly realized that a large part of the definition can be understood to say that a game is a function from a strategy set into a set of lenses, and this part of the game composes by lens composition. The additional data of an open game is a so-called equilibrium relation, which determines which strategies are equilibria in a given situation (each strategy really represents a set of strategies, one for each player). In Reference , the author, Capucci, Gavranovic and Hedges demonstrated that this can be further simplified by realizing an open game as a parametrized map in lenses---a morphism in (in fact, one often wants to consider for, for example, Markov categories , to account for mixed strategies) along with a notion of equilibrium relation defined on the parameter object . We will begin this chapter by recapping this idea. On a conceptual level, there is a natural operation on games, called external choice, which assigns to two open games a new game, whose environment begins by making a choice between one of these games, lets that one happen, then provides some payoff for it at the end. The natural type for the interface of this operation is but unfortunately the category of optics doesn't have coproducts. The category of lenses can be extended with coproducts (into dependent lenses), but the analysis of mixed strategies makes probability an absolute necessity for a useful theory of games. The introduction of Markov fibrations provides a solution to this problem, and in the second half of this chapter, we provide such an external choice operation on open games. Although the approach based on Markov fibrations is novel, the approach to defining the external choice operator is otherwise very similar to one appearing in presently unpublished work by the author, Braithwaite, Hedges and Videla (this paper used a more specialized approach to adding coproducts to ). Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-CA6T/ efr-CA6T /efr-CA6T/ Open games in Monoidal categories Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-EGK8/ efr-EGK8 /efr-EGK8/ Definition Let be a monoidal category. A selection relation on an object is a relation . We write for the statement . Selection relations are ordered by inclusion, and so form a (posetal) category, which we denote . Given we define the pushforward on selection relations by . In other words, if and only if there exists so that and . It is clear that pushforward is monotone, so that this defines a functor Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-YKR3/ efr-YKR3 /efr-YKR3/ Remark Of course, there is an equally good "pull-back" operation on selection relations given by (this is the pushforward in ). These are adjoint representatives of the same profunctor, which should arguably be regarded as the primary object of interest---that is, we could work with a functor , the category of categories and profunctors, where we say two selection relations are related by if . However, we will stick with the pushforward definition for now, since it is conceptually simpler and good enough for our purposes. Eigil Fjeldgren Rischel 2025 5 5 https://erischel.com/efr-BNB2/ efr-BNB2 /efr-BNB2/ Example Let . There is a selection function defined by if and only if is a maximum of . (Identifying maps with points and maps with functions ) In the same category, for each there is a selection function given by . This corresponds to satisficing at the value (Reference )---that is, selecting any strategy which achieves this value or greater. For with semiCartesian, there is a selection function (using the same identification as above). The agents with this selection function are called predicting agents in Reference . These agents attempt to predict the value the environment will return to them. For the fiberwise opposite of manifolds and smooth bundles---that is, the category of lenses between smooth manifolds---there is a selection function on the tangent bundle given by if and only if ---that is, if is a fixpoint of the dynamical system identified by . Let be a dynamical systems theory and work in the category of lenses. Suppose the monoidal structure on is Cartesian, so that a lens is the same as a map . Note that has a distinguished section given by the identity. Then there is a selection function on each object where a lens , given by is in equilibrium with respect to a lens if is a trajectory between those systems---that is, if it is an equilibrium state of the smooth dynamical system . This subsumes the two previous examples. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-PTAE/ efr-PTAE /efr-PTAE/ The Nash Product Definition Let be selection relations. Their Nash product is given by Let us unpack this very dense definition. Given a context , a strategy is in equilibrium if: It decomposes as a tensor product of . Composing with , we obtain a map . This is the context of the first player assuming the second player plays . must be an equilibrium strategy for this map. Simultaneously, must be an equilibrium for the analogous composite of and . Eigil Fjeldgren Rischel 2025 5 5 https://erischel.com/efr-5E1V/ efr-5E1V /efr-5E1V/ Example Consider the selection relation on . A point is an equilibrium for a function if and only if maximizes and maximizes . In other words, if is a Nash equilibrium (Reference ) in the usual sense for the game with payoff matrix . Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-Z2BR/ efr-Z2BR /efr-Z2BR/ Proposition The Nash product along with the map given by the full set equips with a lax monoidal structure. Eigil Fjeldgren Rischel202554Proof It is trivial to verify associativity and unitality of the monoidal structure. The only hard part is to verify that is actually a natural transformation. To that end, let and be morphisms of . Let . First consider the statement . This holds if and only if factors as so that . This in turn means and and analogously for . Now consider the statement . This means that factors as so that and analogously for , which in turn means that factors as so that . Using the axioms of a monoidal category, it is straightforward to see that these two requirements on are equivalent, hence we have our naturality. Eigil Fjeldgren Rischel 2025 5 5 https://erischel.com/efr-UI90/ efr-UI90 /efr-UI90/ Corollary Since is lax monoidal, its Grothendieck construction acquires a monoidal structure (Reference ). We write this category ---explicitly, it is given as follows: The objects are pairs where and is a selection relation on it The morphisms are morphisms so that, for every , The monoidal structure is given by Now we are ready to make the slick definition of open games. Eigil Fjeldgren Rischel 2025 5 5 https://erischel.com/efr-1WGM/ efr-1WGM /efr-1WGM/ Definition Let be a semiCartesian symmetric monoidal category. The symmetric monoidal double category of open games in is . Note that in this case, and . Thus a selection function decides, for each payoff function , which of the states are suitable equilibria. Before we proceed to the case of stochastic lenses, we will pause briefly to make a small modification to the preceding theory as presented in Reference . In a game with forwards play function there are two ways to talk about the player's "choice"---we may say that the player chooses a strategy which then has some effect. Or we may say that the choice is really the and the strategy is the "precommitment" of choosing what to do given each possible . Once we introduce randomness---working in , for example---we see that there are two distinct ways for a player to make a random choice: first, his strategy may be stochastic---that is, he is choosing a random strategy. Or the morphism may be stochastic---this means each strategy contains a specification of how to randomly choose given each possible . In the case where , the distinction is between taking and letting the play function be the identity, and taking and letting the play function be the sampling map which stochastically draws an element from a distribution. We take the view that the latter is the proper presentation of this game---this goes along with the terminology in the classical game theory literature, which would certainly regard a distribution on the set of possible moves as a (mixed) strategy. Having represented our games like this, we may restrict ourselves to considering deterministic maps as strategies. This also fixes the awkwardness in the definition of the Nash product, since now every strategy in decomposes uniquely as a pair of strategies. We quickly modify the preceding definitions to make sense of this. Note that we also modify the reparametrization maps to be deterministic (in the base). Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-6I8U/ efr-6I8U /efr-6I8U/ Open games in a stochastic module Definition Let be a symmetric monoidal stochastic module fibration over the Markov category . Recall that acquires a symmetric monoidal structure. Denote as usual . Note that this is stable under the monoidal product, and acts on via the inclusion. Then the category of open games in is the category . When is a Markov prefibration, we overload the notation by writing . We now introduce the notion of strategic equivalence, which identified two open games if they have the same equilibria for every costate which can actually occur as a result of pasting the game into some larger diagram. Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-K0Y8/ efr-K0Y8 /efr-K0Y8/ Context Definition Let be objects of a symmetric monoidal category . A context for is a tuple . We denote the set of contexts . Given a morphism and a context the costate will be called the induced costate. Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-I1K5/ efr-I1K5 /efr-I1K5/ Remark The preceding definition clearly works in a non-symmetric monoidal category as well. In this case, arguably, a context should be defined to consist of maps . However, the problem with this from our point of view is that it doesn't allow the definition of a costate on given a parametrized map (since one cannot commute the past the ), which is what we're interested in. It is also worth observing that contexts are essentially the same thing as optics . We have not specified an equivalence relation on contexts (we do not need it,) but it is easy to see that for any parametrized map, two contexts that are sliding equivalent give the same induced costate. Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-R46X/ efr-R46X /efr-R46X/ Strategic Equivalence Definition Let be a monoidal stochastic module fibration over a Markov category , and consider a 2-cell . We say this is a strategic equivalence if the underlying map in is an isomorphism, and for every context , the induced costate on has the same equilibria (under this isomorphism) as the composite costate on Note that a game up to strategic equivalence is determined by a relation between strategies and contexts. This brings us closer to Hedges' original definition of open game from Reference . The chief difference is that a game in our sense is prevented from "inspecting" the context for those which are not in the image of , in the sense that whether a given strategy is in equilibrium or not cannot depend on this (since we only see a certain costate on ). Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-HLIX/ efr-HLIX /efr-HLIX/ Proposition Strategic equivalence is compatible with composition and tensor in . Eigil Fjeldgren Rischel202556Proof Let be strategic equivalences. By 2-cell composition there is a map , which we must show is a strategic equivalence. Let be a context in . Now a pair of strategies for are in Nash equilibrium for the costate induced by this context if and only if is in equilibrium for the costate induced by the context where is the play function of , and the analogous condition holds for . But if are equivalences, this is clearly equivalent to asking that be in Nash equilibrium for the costate induced by on . This concludes the proof. Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-U99C/ efr-U99C /efr-U99C/ Definition We denote by the symmetric monoidal category of strategic equivalence classes of open games. Note that this category of open games retains a 2-categorical structure, given by deterministic maps which preserve equilibria in every context. However, we will leave a deeper investigation of this structure for future work. Eigil Fjeldgren Rischel 2025 4 30 https://erischel.com/efr-SREZ/ efr-SREZ /efr-SREZ/ Open Games with External Choice Let be an extensive Markov category. Let be a monoidal stochastic module fibration with Markov structure, which has coproducts which are preserved by the pullbacks. Recall that acquires two monoidal structures: one from dualizing the given monoidal structure on which we simply denote , and one from taking the coCartesian monoidal structure in the fiber (which is Cartesian after taking the fiberwise dual, of course), which we denote . Note that is a Markov category. For the rest of this section, fix like this. Eigil Fjeldgren Rischel 2025 5 5 https://erischel.com/efr-Y2ZN/ efr-Y2ZN /efr-Y2ZN/ Lemma Let be objects in , and let be a morphism in . Then there is a canonical map so that the underlying map is Eigil Fjeldgren Rischel202555Proof The first map in the factorization has a deterministic retract (deleting the component,) and using the coproduct-preservation, the coproduct over and pull back to the same object over . Composing the induced stochastic-Cartesian map and the Cartesian map gives the canonical map we wanted. With the interpretation that is the object indexed over this map simply selects one branch randomly and marginalizes to that coordinate in then includes the returned value into the coproduct. Eigil Fjeldgren Rischel 2025 4 30 https://erischel.com/efr-98DR/ efr-98DR /efr-98DR/ Definition When as above, acquires a monoidal structure which we call external choice, and write , given on objects by the coproduct in and on morphisms by the following formula: Given two open games their external choice is has parameter . The play map is given by The selection relation is given (up to equivalence) as follows: Given a context and a deterministic state , they are in equilibrium if factors over the canonical for some . The factorization being given by ,and being given by , we have The idea behind the external choice operator is that, for all the contexts which can actually occur as a result of pasting a game into a larger string diagram, the map has the given form---that is, the probability of landing in each of the two fibers does not depend on the chosen and the conditional distributions on the component of the fiber depend only on . Hence we need only concern ourselves with which states are equilibria for contexts of this form. The choice of the empty set of equilibria for other contexts is merely a convention. Eigil Fjeldgren Rischel 2025 4 30 https://erischel.com/efr-VB6A/ efr-VB6A /efr-VB6A/ Given a state in a Markov category with coproducts, we say a pair form a pair of conditionals if the copairing is a Bayesian inverse of the map . Given a state in , we say a pair of maps form a pair of conditionals if the underlying maps do. Eigil Fjeldgren Rischel 2025 5 5 https://erischel.com/efr-MLF1/ efr-MLF1 /efr-MLF1/ Theorem As defined above, is a symmetric monoidal structure on . Eigil Fjeldgren Rischel202555Proof The monoidal coherences come from the monoidal structure of and it is trivial to see that they preserve the selection relations. The only nontrivial part is proving that is functorial. Hence let be games. The strategy set of is given by For by . In the base, these are the same object . In the fiber, they are given respectively by and Note the coproducts here are the fiberwise ones. There is an obvious lens from the former to the latter (given by the identity map on the base, and the inclusion of two summands in a fourfold coproduct---note that lenses go backwards in the fiber). Letting be a context, and going through the definitions, it is clear that the resulting contexts for the former game factors as this lens followed by the context for the latter game. In other words, this lens is a reparametrization map between the two games. It suffices to verify it is an equivalence. Unpacking the equivalence relation on , note that (in all the possible contexts,) the signal to does not depend on the action of and vice versa, and so for and . Hence they are in equilibrium if and only if they are in equilibrium in for the given context (conditioned on that branch), and similarly the other two. This proves the desired equivalence. Eigil Fjeldgren Rischel 2025 4 30 https://erischel.com/efr-4CSL/ efr-4CSL /efr-4CSL/ Example Consider the (Grothendieck) fibration , which can be viewed as a Markov fibration. Let be a parameterized lens. Denote by the open game with parameters , underlying parameterized lens , and equilibrium relation given by . Then if is another parameterized lens, we have where by an abuse of notation denotes the paramterized lens given by distributing into the coproduct, then projecting into the relevant factor of the product and applying either or A context for either of these games consists of an element of and a function . The external choice game can be seen as having two players, one who gets to play if the context chooses an element in , who must output an element and optimize (according to his private utility function ), the other playing when the input is in and who must choose an element in . The single argmax game can be seen as a single player, who is constrained to play inside the same "branch" of the game as the input (this constraint is encoded in the lens ), and whose utility function is given by the first players' in the first branch, and the second players' in the second branch.x Eigil Fjeldgren Rischel 2025 5 5 https://erischel.com/efr-BNT1/ efr-BNT1 /efr-BNT1/ Example Let be a game, representing an agent who is optimizing the return value in some sense. Then represents the same agent, whose payoff may now depend on an additional bit (a value in ), but whose decisions may not depend on that bit (his selection function may still depend on its distribution). On the other hand, represents the same situation, but where the player's strategy may depend on the bit---he provides two strategies , one for each possibility. This proves that does not distribute over Eigil Fjeldgren Rischel 2025 4 8 https://erischel.com/efr-ZRUZ/ efr-ZRUZ /efr-ZRUZ/ Categories of stochastic dynamical systems Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-5QC5/ efr-5QC5 /efr-5QC5/ Introduction Given a set of inputs and a set of outputs , there are essentially two make sense of the informal description "finite-state automaton which reads inputs from and produces outputs in ". These are the notions of Mealy machine and Moore machine. Simply put, if the set of states is , a mealy machine is a function whereas a Moore machine is a pair . In other words, in a Moore machine the output does not depend on the current input, but only on previous inputs (through their effect on the state), but in a Mealy machine, the input can be passed through directly. Using the language of categorical systems theory, we can make the following definitions: Eigil Fjeldgren Rischel 2024 5 25 https://erischel.com/efr-0006/ efr-0006 /efr-0006/ Moore Machine Definition In a dynamical systems theory, a Moore machine with state space and interface is a lens . The category of Moore machines with interface is the slice category of the functor over ---that is, a morphism of Moore machines is a morphism so that the obvious triangle commutes. Eigil Fjeldgren Rischel 2024 5 25 https://erischel.com/efr-0005/ efr-0005 /efr-0005/ Mealy Machine Definition In a dynamical systems theory, a Mealy machine with state space and interface consists of a costate lens . The category of Mealy machines with interface is the comma category of the functor over ---that is, a morphism of Mealy machines is so that the obvious triangle commutes. A Moore machine in the sense of Definition is what Myers calls a (open) dynamical system, and they are the central object of study in Reference . Arguably, both Moore machines and Mealy machines deserve the name of "open dynamical system"---the difference is how they interact with the external world. Observe in particular that, if , the categories of Mealy and Moore machines agree, both being equal to the slice of over the unit . In other words, the two notions of closed dynamical system coincide. It is clear that both Mealy and Moore machines, in this sense, are special kinds of parametrized morphism in lenses, namely those parametrized by an object of the form . This leads naturally to the idea that there should be a triple category of morphisms of this type, charts, and lenses. We call the parametrized lenses bisystems since they generalize the two types of machine, Moore and Mealy (but we choose to stick with "system" rather than "machine"). Eigil Fjeldgren Rischel 2024 6 13 https://erischel.com/efr-000E/ efr-000E /efr-000E/ Machine Learning as a bisystem Example It is an observation which goes back at least to Reference , and was more thoroughly developed in Reference , that the process of training a machine learning algorithm by gradient descent can be abstracted in the following way: Wanting to learn a function , we choose a parametrized function , where all these are, in the simplest case, Euclidean spaces We take the backwards derivative of obtaining a lens: For each datum we compute the loss gradient , and combining this with and the current parameter , we get a gradient on the parameter space which we can use to update Thus a machine learning algorithm is a sort of bisystem. Indeed our bisystems are essentially an abstracted version of the learners of Fong--Spivak--Tuyeras. The functoriality of this assignment is the main point of the above-mentioned papers. In this chapter, we will put together the ingredients we have assembled so far and construct a triple category of stochastic dynamical systems. We will also give the construction of triple categories of systems in the ordinary case. Note that the theory of Markov fibrations does not quite generalize the ordinary theory of fibrations---only fibrations with a Cartesian base (Example and Proposition ). Since the pullbacks in play such a key role in the theory, it is not clear that this can be dispensed with. Although Cartesian bases certainly cover the vast majority of examples from the literature on categorical systems, it is of course worth noting that they are not a requirement. Moreover, we will see that the construction of the double category of lenses and charts encounters certain problems for a general Markov fibration not seen for ordinary fibrations. Hence we will give a separate description of the triple categories in each of the two cases. The notions of Mealy and Moore machine are both quite old, going back to Reference , Reference . While we do obtain finite-state automata of these types as special cases, our interest is primarily in the analysis of dynamical systems, which tends to ask rather different questions than automata theory. Thus, despite using the terminology, we will not be particularly interested in the actual theory of Mealy and Moore machines. We do mention one recent paper, Reference , which has a category-theoretic approach similar in spirit to our own. Their category of Mealy machines can be obtained, not as our category of Mealy machines above, but by considering maps in the case where are trivial in the secondary component. In the discrete case, such a map is given by . It should also be noted that the idea of embedding Mealy machines as the morphisms is not original, but was communicated to the author by Matteo Capucci. It seems not to have appeared in the literature so far. The idea that there "should" be a triple category of systems, like the one we will construct, has also circulated as folklore, although again an explicit construction has yet to appear. Very similar ideas appear in the work of Shapiro and Spivak, see for example Reference . We begin this chapter with a treatment of double categories of charts and lenses in the context of Markov fibrations (and stochastic modules). The construction does not work quite as well as in the classical case---the difficulty is essentially that the equivalence relation which defines stochastic charts has a directed nature, and given a 2-cell defined in an obvious way and an equation there is not (apparently) in general a way to lift this to an arrow (with ). However, we can construct a double category whose globular horizontal 2-category have connected components given by the stochastic charts (or lenses). We proceed to give an account of categorical systems theory for these double categories. The chief problem posed by the above is that we may not have any good clock systems (Example ). Two equivalent charts should represent the same system, but they may receive different sets of maps from the supposed clock system (and thus have different sets of trajectories). We resolve this by proving that, for lenses with a deterministic base, every equivalence class of lenses has an initial representative. Moreover, mapping out of this initial representative to a representative of some other lens, a 2-cell exists filling a given square if and only if it commutes as a map of lenses and charts in the classical case (recall that over deterministic bases, Markov fibrations become ordinary fibrations). In particular the trajectories of a system with respect to such a clock system depend only on their equivalence class. We follow this up by constructing the above discussed triple categories of "bimachines", that is systems which combine Mealy and Moore machines. We do this both for stochastic and the ordinary case---the procedure is exactly the same, but of course they are different objects, neither generalizing the other. We end by constructing a stochastic dynamical systems theory for smooth dynamical systems---requiring a brief detour to construct a suitable Markov category of smooth kernels. Eigil Fjeldgren Rischel 2025 4 24 https://erischel.com/efr-SIE7/ efr-SIE7 /efr-SIE7/ Double categories of stochastic charts and lenses To construct the double category of charts and lenses for an ordinary fibration , one can use the following procedure: Form the square double category Take the fiberwise opposite of these objects: . Observe that fiberwise opposite preserves pullbacks, and hence this is again a double category. Here we used the following result: If is a Grothendieck fibration and is any category, then is again a fibration (which classifies the lax limits of the composite ), and a natural transformation is Cartesian iff it is levelwise Cartesian. It would be neat to obtain a similar result for Markov fibrations. The first problem with this is that does not generally inherit a Markov structure from . As we noted when we introduced diagram Markov categories, one has to consider the category of deterministic diagrams instead. First, we will see that this indeed works for Markov prefibrations. This implies that lifts from fibrations to stochastic modules. Eigil Fjeldgren Rischel 2025 4 8 https://erischel.com/efr-6G7N/ efr-6G7N /efr-6G7N/ Proposition Let be a Markov prefibration. Recall that by we denote the category of deterministic arrows in . Let denote the ordinary arrow category. Let now denote the category consisting of those arrows in which lie over a deterministic base (but again, where the morphisms consist of commutative squares whose other sides do not necessarily have deterministic bases). Then is a Markov prefibration. This yields a functor , so that . This equation induces a natural transformation , which in turns gives a lift of to the category of stochastic module fibrations, where the induced algebra structure acts pointwise. Eigil Fjeldgren Rischel202548Proof Noting that is a Markov category with the "pointwise" structure, and the deterministic maps consist precisely of the pointwise deterministic maps, clearly , and fibrations are stable under the formation of arrow categories, with Cartesian maps formed pointwise. It is not trivial that this is a prefibration, because given a triangle in ---a "prism"---and a Cartesian lifting, we only know that the maps "at the ends" are Cartesian, not the maps between the triangles. Therefore we can not immediately apply the unique lifting property to say that the square between the induced lifts commute, given some map over . However, by taking the pullback on both sides (and noting that pullbacks are functorial,) we can factor this square into two which live entirely over a deterministic base, and where the Cartesian property therefore imply commutativity. We have already argued that this commutes with restriction to the deterministic part. The natural transformation is induced for completely abstract reasons, by applying to the unit to obtain a map which by the universal property of induces the desired map . Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-DXJR/ efr-DXJR /efr-DXJR/ Corollary Let be a small category, and let be a Markov prefibration. Let . Then the functor is a Markov prefibration. Eigil Fjeldgren Rischel202556Proof Note that is a limit of the categories and prefibrations are stable under these limits. The question is now If is a Markov fibration, we get a stochastic module structure on ---does it present a markov fibration? There is an induced map (where the latter is taken by convention to mean the full subcategory of functors whose image in consists of deterministic arrows). Is this an isomorphism? (If it is, clearly this implies point 1) Unfortunately it's not clear that either of these are true---the surjectivity of cannot a priori be lifted to the arrow category. The issue is that, given a map in consisting of, say it is not sufficient to find charts representing each of these---we must find a chart of squares representing the square. This is not guaranteed by the Markov fibration structure, and a similar issue comes into play for the equivalence witnesses. We may attempt to ignore this issue and simply try to form a double category given a stochastic module , but here the problem is that does not commute with limits in general. Hence we can not easily define a composition on the 2-cells of lenses obtained this way. There are various ways we might attempt to remedy this problem. One approach would be to formulate a behavioural notion of "commutativity" for squares of stochastic lenses and charts, but the problem with this is that it is not obvious whether this property is stable under composition. The basic problem stems from the fact that chart equivalences have a "directed" nature, and given a morphism of precharts and an equivalence (for example given by a stochastic section satisfying suitable conditions), there is not in general a way to lift this back into an equivalent with a map to . This observation leads to the idea that we might define a double category of precharts which has the directed equivalences among its morphisms (going only in one direction). We will begin by constructing this double category. Eigil Fjeldgren Rischel 2025 4 23 https://erischel.com/efr-KO4T/ efr-KO4T /efr-KO4T/ Definition Let be a (Grothendieck) fibration, and let be a faithful, identity-on-objects functor. Suppose admits pullbacks, and given a pair of morphisms over , where is in , there is a unique common factorization . The double category has as the vertical category, spans in equipped with a section as horizontal cells, and maps of spans which commute with the sections as 2-cells. The double category lying over has as the vertical category, and spans where is Cartesian, decorated with a section in as the horizontal cells, and maps of such spans (so that the underlying thing commutes with the sections) as the 2-cells. We will call the horizontal cells decorated spans. There is an apparent forgetful functor To spell it out, a 2-cell in consists of a diagram of this form in , where are Cartesian, and (writing for the object underlying , and so on) two sections of the underlying maps in , so that the second diagram also commutes in : Naturally, we are interested in the case of for a stochastic module . Then the decorated spans are representatives for stochastic charts. We will start by introducing a loosed notion of 2-cell for these spans, which combines the directed equivalences with the ordinary deterministic 2-cells of spans. Eigil Fjeldgren Rischel 2025 4 24 https://erischel.com/efr-RD28/ efr-RD28 /efr-RD28/ Definition Let be a Markov category and let be a stochastic module over . Let be decorated spans in , and let be morphisms in , so that we have a square A 2-cell of decorated spans for this data consists of a morphism (that is, possibly stochastic), satisfying the following two conditions. First, the diagram in the base must commute. Given this, there is an induced square in , where the bottom map is given by pulling back along , in the sense of Lemma . The second condition is that this square must also commute. Eigil Fjeldgren Rischel 2025 4 24 https://erischel.com/efr-DRU6/ efr-DRU6 /efr-DRU6/ Lemma 2-cells of decorated spans compose---that is, given a diagram where the horizontal maps are decorated spans, and the vertical maps are maps in with deterministic base, and given maps of decorated spans there is a map of decorated spans Eigil Fjeldgren Rischel2025424Proof We can easily compose the two maps to get . Now, the question is whether the perimeter of this diagram commutes: Note that the top square is commutative by assumption, since pullbacks compose (even along stochastic maps) and this is the assumption that is a cell. The bottom square is the result of pulling back a commutative square over again along . Note that pullback along stochastic morphisms is not in general functorial---but since the vertical parts of this square are themselves pulled back from , this composition is preserved by pullback along . This finishes the proof. Eigil Fjeldgren Rischel 2025 4 24 https://erischel.com/efr-L7X0/ efr-L7X0 /efr-L7X0/ Proposition 2-cells of decorated spans compose horizontally: Given a square where the horizontal maps are precharts and the vertical maps are morphisms in , and given and , there is a prechart morphism . Eigil Fjeldgren Rischel2025424Proof Unlike the proof of Lemma , this is straightforward: If the carrier of is , and of (for ), then by definition the composites are carried by the pullback . There is a canonical map over , given by the independent pairing of and . Since pullbacks compose, the square over that must commute is given by the two commutative squares induced by , pulled back and composed with each other. Here we are pulling back along the deterministic projections from the pullback, and hence these commutative squares are preserved, and hence the composite square commutes as well. Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-MUNG/ efr-MUNG /efr-MUNG/ Corollary Let be a stochastic module over . There is a double category which has decorated spans as its horizontal maps, morphisms in as its vertical maps, and decorated span 2-cells as its 2-cells. Moreover, there is another double category which has decorated spans in as its horizontal maps instead. (The modification of everything above to lenses instead of charts is obvious). In fact, the globular 2-cells are in a sense exactly the equations defining the set of charts: Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-CU3J/ efr-CU3J /efr-CU3J/ Lemma Let be parallel decorated spans in (or ). They represent the same stochastic chart (lens) if and only if there exists a zig-zag of globular 2-cells between them. Eigil Fjeldgren Rischel202556Proof Unwinding the definitions, a 2-cell is precisely a map exhibiting the equality of the two decorated spans, as in Lemma The following straightforward lemma shows that charts over deterministic bases have initial representatives as spans. This will be important later: Eigil Fjeldgren Rischel 2025 4 24 https://erischel.com/efr-10CM/ efr-10CM /efr-10CM/ Proposition Suppose given a square in (or ) Suppose further the underlying square in is deterministic. Then: If there exists a filling decorated span 2-cell, the image in commutes. If is the carrier of and the left leg is an isomorphism, then this implication is an equivalence. Eigil Fjeldgren Rischel2025424Proof Given a stochastic chart carried by so that is deterministic, recall that we can first pull back to the equalizer of the two maps , then along the prescribed section . Note that this gives a 2-cell from this canonical representative with carrier to the initial representative. Given two such cells, we get a square But as part of the square surrounding the 2-cell, there is given a map , which must make this square commute. The functoriality of base change (pulling back the map ) proves this bottom map is again a 2-cell. But the property for this map to be a 2-cell is exactly the property for this square to be a commutative square in . This proves both parts of the statement. Eigil Fjeldgren Rischel 2025 4 8 https://erischel.com/efr-7DV5/ efr-7DV5 /efr-7DV5/ Double Categories of Stochastic Dynamical System Eigil Fjeldgren Rischel 2025 4 25 https://erischel.com/efr-HS7G/ efr-HS7G /efr-HS7G/ Stochastic Dynamical Systems Theories Definition A stochastic dynamical systems theory consists of a stochastic module over equipped with a section of the underlying fibration. We adopt the notation for the double category which was denoted above. We use the symbol for the horizontal (span) morphisms and for the vertical morphisms (in ). Given a stochastic dynamical systems theory , the double category of dynamical systems has Objects triples Horizontal morphisms given by a pair where is a square filling Vertical morphisms given by a pair where is a square filling Eigil Fjeldgren Rischel 2025 4 25 https://erischel.com/efr-N5V5/ efr-N5V5 /efr-N5V5/ Proposition Let be a stochastic dynamical systems theory. Then is an ordinary dynamical systems theory. There is a double functor . This functor is full on vertical morphisms, and on 2-cells. Let denote the subcategory spanned by systems with deterministic readout, prelenses with deterministic base, and all the morphisms of . Then the restricted functor admits a chartwise right adjoint, which assigns to each system or prelens its equivalence class. Eigil Fjeldgren Rischel2025425Proof The functor simply acts as the functor in Proposition . Since that inclusion is full on 2-cells, this one is full on vertical morphisms, and since a 2-cell in is merely a 2-cell in between specific objects, it is also full on 2-cells. The right adjoint property likewise follows from the analogous property of the inclusion functor on arenas. Eigil Fjeldgren Rischel 2025 4 25 https://erischel.com/efr-JQI9/ efr-JQI9 /efr-JQI9/ Proposition Let be a Markov prefibration. Then there is a double functor which acts as identity on the morphisms of , and carries each lens to the prelens . This double functor is full on 2-cells. The restriction to admits a right adjoint, which carries every prelens to its equivalence class. Eigil Fjeldgren Rischel2025425Proof First we must verify functoriality with respect to lens composition. To that end, let be deterministic morphisms and be lenses over them. Their composite as prelenses has carrier and as the morphism in the fiber, which is exactly the prelens associated to their composite as lenses. Second, we must verify fullness on 2-cells. But this is a special case of Proposition . Finally, Proposition is precisely the statement that assigning a prelens to its equivalence class lens is right adjoint to this (and in particular that it forms a functor) Recall that in Myers' categorical dynamical systems theory, trajectories of a system are identified with chart morphisms from a "clock" system---thus for example trajectories of a smooth dynamical system are exactly those maps which, when is equipped with the vectorfield , are homomorphisms. In general this presents an issue for our replacement category ---since we wish to regard two systems given by equivalent lenses as equivalent, but their set of homomorphisms from a given clock system is not necessarily in bijection. In the general case, we do not presently have a way around this problem---but at least for clock systems with deterministic readout, the above presents a solution: choosing the initial representative for such a system, we find that the set of trajectories does not depend on the equivalence class of the target system. Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-MWCE/ efr-MWCE /efr-MWCE/ The triple category of dynamical systems Having done the prep-work in § (and § ), we can jump directly into the construction: Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-EPI9/ efr-EPI9 /efr-EPI9/ Definition Let be a symmetric monoidal dynamical systems theory. Then this diagram: depicts two strict double categories, each with a symmetric monoidal structure, and a strict double functor between them which is (non-strictly) a symmetric monoidal functor. This induces an object of . Applying under we obtain a symmetric pseudomonoid in internal pseudocategories in . Denote by this induced object. Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-WX1V/ efr-WX1V /efr-WX1V/ The Structure of Remark has the following structure: The objects are simply the objects of ---the bundles. There are three types of 1-cell: Lenses, that is morphism in , which we write , charts, that is morphisms in , which we write , and bisystems, which are pairs , and which we write . Moreover there are three types of 2-cell: Lens-chart cells, which are the 2-cells of . Note that this double category is thin. We will say a square of lenses and charts commutes if it is filled by such a 2-cell. Chart-bisystem cells---given charts and systems a square filling this is a choice of map so that the resulting lens-chart square commutes. Lens-bisystem cells. Given lenses and systems a 2-cell consists of an isomorphism so that the resulting square of lenses commutes (recall that isomorphism charts are the same as isomorphism lenses). The charts, bisystems and chart-bisystem cells form a pseudo double category with the obvious composition. So do the lenses, bisystems and lens-bisystem cells. Finally, there is a notion of 3-cell given by a box whose sides are 2-cells of each kind, so that the resulting diagram in (with isomorphisms on two sides) commutes. The lens-bisystem double category is (recall that ), with the action given by the functor (restricted to isomorphisms). The chart-bisystem double category is the result of taking , a pseudocategory in double categories, and applying where takes the horizontal category of a (strict) double category. The 3-cells, of course, are the 2-cells of . Analogously to the above, for each class of 1-cells, there is a double category with those as the objects, the two types of cell as the two morphisms, and the 3-cells as the 2-cells. All of these six double categories admit a symmetric monoidal structure. We can recover the ordinary category of (Moore) systems as the slice over , in the following sense: Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-2M0M/ efr-2M0M /efr-2M0M/ Proposition There is a pullback diagram in In a double category , there is a "horizontal slice" 1-category having objects the horizontal maps and morphisms given by 2-cells that are identity on the left boundary, composed vertically. Similarly there is a "vertical slice". This is given by a similar pullback in ---we simply do this one level up. Note that Myers' construction of the double fibration in fact uses the vertical slice in this sense. This somewhat trivial observation means that any composition in which produces 2-cell under in fact produces a morphism of systems in the ordinary sense. Replacing with another object, we may regard the slices as further-parametrized versions of . There is essentially no difficulty in applying this to the Markov case: Eigil Fjeldgren Rischel 2025 6 20 https://erischel.com/efr-W413/ efr-W413 /efr-W413/ Definition Let be a stochastic dynamical systems theory. Consider the double category whose vertical category is and horizontal category is , with commutative squares as the 2-cells (this is the transpose of ). This carries an obvious symmetric monoidal structure, and acts on the double category of stochastic arenas via the functor . Denote by the symmetric monoidal triple category induced as in Definition by this data. The information contained in is much as above, although the complications involved in the double category of stochastic arenas remain present. Our bisystems are reminiscent of the energy-driven systems of Capucci, Lynch, and Spivak (Reference ). Indeed, their is essentially the bisystems in the (ordinary) doctrine of smooth dynamical systems. As we mentioned in the introduction, Shapiro and Spivak (Reference ) have previous developed a structure which consists of the bisystems for the discrete dynamical systems doctrine (i.e ). has a tremendous amount of structure coming from the representation of lenses in this doctrine as the category of polynomial functors, which can't be replicated for a general systems theory (and certainly not for a general stochastic systems theory). Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-9J8G/ efr-9J8G /efr-9J8G/ A Stochastic Dynamical Systems Theory of Smooth Manifolds In this section, as the title suggests, we construct a stochastic dynamical systems theory of smooth manifolds, with the usual tangent bundle. The main point is to construct a Markov category containing the smooth manifolds which is pullback-positive. We do this by considering the larger category of diffeological spaces. In order to make the topology work, we need to complicate the notion of diffeological space a bit, but having done so, we obtain a representable Markov category which is pullback-positive, and contains as a full subcategory of the deterministic maps. A kernel is a Markov kernel valued in Radon measures which is weakly continuous---so induces a linear map taking to the function on the spaces of continuous functions---and which furthermore smooth in the sense that this operation preserves the smooth functions. This Markov category of "smooth stochastic maps" may be of some independent interest. Eigil Fjeldgren Rischel 2025 5 2 https://erischel.com/efr-CBBU/ efr-CBBU /efr-CBBU/ Definition Let denote the full subcategory of spanned by the objects for each . Note that has finite products, and is generated by the object under finite products. Eigil Fjeldgren Rischel 2025 5 2 https://erischel.com/efr-E1SZ/ efr-E1SZ /efr-E1SZ/ Diffeological Space Definition A smooth space is a sheaf on in the standard topology of open covers. A smooth space is a diffeological space if, for each the map is injective. A diffeological space with underlying set is called a diffeology on , and consists of specifying which maps are smooth. We call these maps smooth plots. Given a subset there is an obvious canonical diffeology on given by taking the plots to be those functions whose image in is smooth. We call this the subspace diffeology. A morphism of diffeological spaces is called a smooth map. It is equivalently a function which carries smooth plots to smooth plots. Eigil Fjeldgren Rischel 2025 5 2 https://erischel.com/efr-DX6S/ efr-DX6S /efr-DX6S/ Diffeo-Topological space Definition A diffeo-topological space is a diffeological space equipped with a topology so that all the smooth plots are continuous. A map of diffeo-topological spaces is a smooth map (for the diffeology) which is also continuous (for the topology). Given a subset there is an obvious canonical diffeo-topology on given by taking the plots to be those functions whose image in is smooth, and equipping with the subspace topology. Eigil Fjeldgren Rischel 2025 5 2 https://erischel.com/efr-42SC/ efr-42SC /efr-42SC/ Lemma The category of diffeo-topological spaces admits all limits, given by taking the limits in topological spaces and diffeological spaces (which have the same underlying set). Eigil Fjeldgren Rischel 2025 5 2 https://erischel.com/efr-7RS2/ efr-7RS2 /efr-7RS2/ Proposition Let be a diffeo-topological space whose underlying space is Tychonoff. Then the space of probability measures has a canonical diffeology given by those plots so that for each continuous function on the resulting map is continuous, and if is smooth, then this is smooth as well. With this diffeology, and the topology of weak convergence, is a diffeo-topological space. This defines a commutative affine monad on the category of such diffeo-topological spaces. Eigil Fjeldgren Rischel202552Proof The topology of weak convergence on is such that is continuous if and only if the expectation map carries continuous functions to continuous functions. But this is part of the requirement to be a smooth plot, so certainly this is a diffeo-topological space. Since the linear operator associated to under the unit is merely evaluation at the unit is clearly smooth. Consider . To test this is smooth, let be a plot. We must show is a plot. So let be a continuous (resp. smooth) function. We must show its expectation a is continuous (resp. smooth) function of . By construction is continuous (resp. smooth), and so since is a plot, the map is continuous (resp. smooth). But this is exactly what we wanted. The monad laws follow from their holding in . Since a commutative monad is equivalently a strong monad satisfying a certain equation (which holds for this monad in and therefore also here), it suffices to show that the strength is smooth. This follows by a completely analogous argument. Eigil Fjeldgren Rischel 2025 5 2 https://erischel.com/efr-S87B/ efr-S87B /efr-S87B/ Corollary The category of Tychonoff diffeological spaces and Kleisli maps for the monad described in Proposition is a pullback-positive Markov category. Its deterministic category is . There is a fully faithful functor which preserves transverse pullbacks. The maps between smooth manifolds are given by weakly continuous families of Radon probability measures, so that the expectation operator carries smooth maps to smooth maps. Eigil Fjeldgren Rischel 2025 5 2 https://erischel.com/efr-LPX2/ efr-LPX2 /efr-LPX2/ Remark Since we describe properties of kernels in terms of their corresponding linear operator on function spaces, it would seem natural to take the function spaces as the basic object. Hence we might consider the category -algebras with some relaxed notion of maps between them. The tricky part there is to find some reasonable class of maps so that the tensor product (coproduct) of -algebras extends to these. (Since it is not the same as the tensor product of -algebras, linear maps do not automatically extend to the tensor product). In particular, when considering kernels where is a "fat point of order 1"---that is, functions on have a value at the point and a derivative---it is not clear what sort of continuity condition the derivative operation on should satisfy, nor how to define this in a general way for all -algebras. It would certainly be of interest to synthetic computational geometry to have such a Markov category, but we leave this for future work. We will now give an example of how to represent the training dynamics of a machine learning system using the tools developed so far. As discussed previously, given a parameterized function , its reverse derivative naturally becomes a parameterized lens, and the composition of these describe how gradient vectors are passed around to compute an update during training. It is natural to want to compose this lens with the data-generating distribution , (along with some more context describing the loss function, etc) to obtain the training dynamics of such a model. This requires a category of parameterized lenses which allows stochastic maps in the base. The goal of combining this feature with non-trivial tangent bundles was one of the original motivations for developing a theory of stochastic lenses. Eigil Fjeldgren Rischel 2025 5 3 https://erischel.com/efr-IMZ3/ efr-IMZ3 /efr-IMZ3/ Example Consider the Markov prefibration . Equip this with the section - this described discrete-time systems (whose update is required to be smooth in the input and present state). This is clearly a symmetric monoidal functor and thus defines a systems theory. Note that this is completely different from the ordinary tangent bundle, despite the coincidence of notation. Let and be two bisystems in this theory. As in § , we may define a parameterized lens , denoting by the coproduct in lenses, and by the Markov structure defined in Corollary . Observe that is simply the indexed set . There is an obvious indexed map from this to , given by and . In words, we receive an update either to the -state or to the -state. We apply this update to the relevant state and leave the other alone. This defines a lens , which we may compose with the above to obtain a bisystem . Let us denote this by . This has very much the same flavor as the external choice for open games, although we will not develop the theory of this operation in detail here. Now, given a smooth (deterministic) map , where are smooth manifolds (not merely diffeo-toplogical spaces), we obtain a lens , where denotes the cotangent bundle. (This does not, prima facie, make sense for a general diffeo-topological space). Let us take as given some family of lenses for various . Such an operation amounts to choosing a way of updating given a cotangent vector---hence we can see it as an optimization algorithm. One example of such would be gradient descent, which given a choice of Riemann structure on , takes a step of a given length in the direction which most quickly decreases the given covector. (It should be noted that there are more complicated optimization strategies which don't fit this particular pattern - for example, momentum algorithms have to maintain some extra internal state other than . But let's stick with this pattern for this example). Note also that we're not assuming the optimizers are a natural transformation or anything like that. Now we are ready to build the neural network architecture known as a generative adversarial network, or GAN (Reference ). Let us first describe the idea. Our goal is to generate additional samples from some distribution, given a set of existing samples---for example, our goal may be to generate more pictures in the same style as an existing corpus. Suppose our data is of type , and let be the data distribution. We fix some latent distribution , where is any space of our choice---usually, is and is a Gaussian. Finally we choose two neural networks, the generator , and the discriminator . The goal of the discriminator is to discriminate real samples from the data from generated samples, by providing a low value on the generated samples and a high value on the true samples. The training process now goes as follows: for each step of training, we either sample from the latent distribution, and have the generator use this to generate a sample, or draw a sample from the data distribution (choosing between these with some probability ). Then in either case, we have the discriminator score the generated sample. If the sample was generated by the generator, the discriminator's loss is equal to its output, otherwise it is equal to times its output. We update the discriminator according to the gradient of this loss (minimizing it), and update the generator (if we are in the branch where it was run) according to the negative of the gradient of this loss with respect to the generator parameters---this amounts to doing a gradient descent update on the generator for the negative of the discriminator loss. We can represent this schematically using the following tape diagram (see Reference ): ] \tikzstyle {Tape boundary}=[-, draw={rgb,255: red,191; green,0; blue,64}] \resizebox {\textwidth }{!}{ \begin {tikzpicture}[scale=0.5] \begin {pgfonlayer}{nodelayer} \node [style=State] (0) at (-5.5, -2.5) {Data}; \node [style=State] (2) at (-10, 5.25) {Latent}; \node [style=System] (4) at (-4.5, 5.25) {Generator}; \node [style=State] (6) at (-4.5, 3) {+1}; \node [style=State] (7) at (-5.5, -5.25) {-1}; \node [style=System] (8) at (4.75, 0.25) {}; \node [style=System] (9) at (4.75, 0.25) {Discriminator}; \node [style=System] (10) at (9.25, 0.25) {Loss}; \node [style=none] (11) at (-3.75, 5) {}; \node [style=none] (12) at (-1.75, 4.75) {}; \node [style=none] (13) at (1.5, 0.25) {}; \node [style=none] (14) at (3.25, 0.5) {}; \node [style=none] (15) at (5, 0) {}; \node [style=none] (16) at (5, 0.5) {}; \node [style=none] (17) at (9, 0) {}; \node [style=none] (18) at (9, 0.5) {}; \node [style=none] (19) at (6.5, -1) {}; 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\node [style=none] (69) at (13, -2.25) {}; \node [style=none] (70) at (13, 1.75) {}; \node [style=none] (71) at (-15, 1.5) {}; \node [style=none] (72) at (-15, -1.5) {}; \node [style=State] (73) at (-16, 0) {Branch}; \node [style=ground] (74) at (-2.75, -1.5) {}; \node [style=none] (75) at (-1.25, -1) {}; \node [style=none] (76) at (-19.25, -1.5) {}; \node [style=none] (77) at (-19.25, 1.5) {}; \node [style=ground] (78) at (-7, 5.5) {}; \end {pgfonlayer} \begin {pgfonlayer}{edgelayer} \draw [style=Arrow] (15.center) to (17.center); \draw [style=Arrow] (18.center) to (16.center); \draw [bend left=45, looseness=0.75] (6) to (20.center); \draw [bend right, looseness=1.50] (7) to (20.center); \draw [bend left, looseness=1.25] (11.center) to (21.center); \draw [bend right, looseness=1.25] (0) to (21.center); \draw [style=Arrow] (21.center) to (13.center); \draw [style=Arrow] (20.center) to (19.center); \draw [style=Arrow, bend right, looseness=1.25] (22.center) to (12.center); \draw (22.center) to (14.center); 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\draw (32.center) to (78); \end {pgfonlayer} \end {tikzpicture}} ]]> The two "tapes" branching off at the start represent two maps (bisystems) composed by , while the backwards wires indicate the flow of the gradients. The ground symbol indicates a value being discarded. Note that the theory of tape diagrams has only been developed formally for distributive categories, and for essentilly the same reason as in § , we do not have distributivity in this case. However, the interpretation of this diagram is still unambiguous---distributivity is required to make tape diagrams complete, not to make them sound. Concretely, if we tried to represent the tensoring of this system with another system, we would have no way of doing it, but if the category was distributive we could do so by adding this additional system to each of the branches. Still, the figure is best viewed as a visual aid rather than a formal representation. Backlinks Related Eigil Fjeldgren Rischel 2025 6 20 https://erischel.com/efr-RUTY/ efr-RUTY /efr-RUTY/ Generative Adversarial Network Example The idea of a generative adversarial network, or GAN (Reference ), is to generate synthetic data which appears to be drawn from the same distribution as a given dataset. The basic idea is to train two networks in parallel: a discriminator, which is trained to distinguish the generator's output from the real dataset, and a generator, which is trained to fool the discriminator. In each training step, one of two things happen: We take a sample from the dataset, have the discriminator score it, and update it according to gradient descent to maximize the score. We take a sample from the latent distribution, run the generator on it, then run the discriminator on the output. Then we update the discriminator with gradient descent to minimize the score, and update the generator with gradient descent to maximize the score in this case. A GAN is naturally thought of as a sort of game played between two players, and hence has been an interesting example to study the parallels between the algebraic structure of machine learning and open games. However, since the game is essentially probabilistic, modeling this requires a category of lenses which contains both smooth manifolds, their tangent bundles and derivatives, as well as a good supply of stochastic maps. Additionally, the most natural way to model a GAN is as a morphism on a coproduct of spaces - hence we want coproducts in the lens category as well. The systems doctrine constructed in this section has all these features. Fix a distribution on , the data distribution. Typically this will be a discrete distribution given by a dataset, perhaps of images. Also fix a latent distribution on . Typically this will be something like an uncorrelated Gaussian. Choose a generator network and a discriminator network . In practice these are given by neural networks, but for now we require them simply to be smooth maps. (Actually, neural networks with ReLU activations are not smooth, but we ignore this subtlety for now) Finally choose optimization algorithms and for the generator and discriminator networks. Given this data, we obtain a bisystem with parameter space as follows: First, there is a morphism given by . Applying to this, and composing with the optimization algorithm for the generator we obtain a bisystem with parameter space . Call this Second, there is an obvious morphism which simply runs the discriminator in both cases. Taking the reverse derivative of this, and applying the optimization algorithm for the generator, we obtain a bisystem . Call this Now there is an endomorphism lens which is the identity in the forwards direction, and multiplication by in the backwards direction. Call this . The composite is now a bisystem . Finally, we compose this with the covector field given by in the first branch, and by in the second branch. The semantics of this when given a data point is to run the discriminator, then update according to the gradient of the output---that is, to maximize the output. Given a latent sample , we first run the generator, then the discriminator. The discriminator is now updated in the opposite direction---to minimize the output on this value, while the generator is updated in the positive direction---to maximize the value of its generation on this sample. So far we have worked inside the doctrine of smooth dynamical systems. However, passing into the stochastic doctrine just described, we may precompose these bisystems with the map given by sampling from with probability and from with probability . The resulting dynamical system describes the training dynamics of the GAN. (One can also consider more complicated versions of this, where the number is controlled by another dynamical system. One can also embed a controller which weighs the gradients sent to the two networks, so that for example if the discriminator is too good, one slows down its learning rate compared to the generator for a while.) Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-POD2/ efr-POD2 /efr-POD2/ Preliminaries Eigil Fjeldgren Rischel 2025 5 6 Introduction We will begin by reviewing some theory that plays a key role in this thesis. With a very few exceptions, nothing here is novel, but we find it useful to include these here---both for the convenience of the reader, to familiarize them with theory that we will make constant reference to, but also to set the stage for our contributions. First, the theory of (Grothendieck) fibrations. There is far too much to say about these for such a brief space, so we will limit ourselves to what we need for the rest of the thesis, especially for the section on Markov fibrations. Second, we will give an overview of optics and lenses, in a bit more detail than the introduction. We have already given a review of the sources there, but we will find it useful to put this on proper footing. Next, we will review the theory of Markov categories. This is a synthetic approach to probability theory, introduced by Fritz Reference , and since developed further by many collaborators, including the author. Here we note the only exceptions to the claim that nothing in this chapter is novel. First, the notion of representable Markov category Definition was introduced by Fritz, Gonda, Perrone, and the author in Reference . We will not give a thorough treatment here, but since representable Markov categories are so ubiquitous, we will frequently note how different properties or structure on a Markov category relates to representability. We will also introduce a few new concepts which play a role in the theory of Markov fibrations in § . These are of no great independent interest, as far as we can tell, nor are they difficult, but this seemed the best place to put them. Finally, we give a brief review of Myers' categorical systems theory. This will mainly be to set the stage for § , where we develop a triple categorical version of the theory. Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-3AEA/ efr-3AEA /efr-3AEA/ Fibrations In category theory, there are many families of categories indexed by the objects of some other category. For example, for each commutative ring, we have the category . Given a ring homomorphism there is an induced restriction of scalars functor (given simply by composing the module structure by ), and this is (contravariant) functorial, assembling into a functor . In most cases, one can not expect strict functoriality as above. From an abstract point of view, it makes sense that one should really ask only for a natural isomorphism , up to some coherence conditions. This assembles into a so-called pseudofunctor into the 2-category . From a concrete point of view, there are many natural families of categories which arise as pseudofunctors. For example, restriction of scalars always has a left adjoint (extension of scalars, given by viewing as an -module via the map )---since adjoints compose (that is, if and then ) this must be functorial up to natural isomorphism, but this is the best we can promise. To avoid the higher categorical algebra involved in working with pseudofunctors, Grothendieck introduced the notion of fibration in Reference . Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-Z7EF/ efr-Z7EF /efr-Z7EF/ Definition Let be a functor. Given , write for the (strict) pullback . Explicitly, this consists of the objects in with and the morphisms with . Let be a morphism. A map with is locally Cartesian if for each with , postcomposition with induces a bijection A map is Cartesian if for every and with there is a bijection note that every Cartesian map is locally Cartesian (take ) is a Grothendieck fibration (or just fibration) if, for every such that , there exists a Cartesian map (for some ) so that Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-Z4L4/ efr-Z4L4 /efr-Z4L4/ Lemma is a Grothendieck fibration if and only if every admits a locally Cartesian lift, and the class of locally Cartesian morphisms in is stable under composition. Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-QHXA/ efr-QHXA /efr-QHXA/ Theorem Let be a Grothendieck fibration. For every select a Cartesian lift of . Then there is a unique extension of to a functor so that the squares commute. With this, the assignment assembles into a pseudofunctor . Eigil Fjeldgren Rischel 2025 5 5 https://erischel.com/efr-WKG8/ efr-WKG8 /efr-WKG8/ Remark If is such that each morphism admits merely a locally Cartesian lift (but these do not compose,) it is called a prefibration. Note that this is unrelated to our notion of Markov prefibration (reading ahead a bit, the "Cartesian" maps in a Markov prefibration do compose, but they enjoy the unique lifting property only for a subset of morphisms). This clash of terminology is perhaps unfortunate, but other potential prefixes seemed inferior (quasi-, pseudo-, semi-). Eigil Fjeldgren Rischel 2025 5 7 https://erischel.com/efr-D7W0/ efr-D7W0 /efr-D7W0/ Example Let be any category, and let denote the arrow category. Then the codomain functor is a fibration if and only if admits all pullbacks, and in this case the functors given are given by pullback along . The functors are sometimes referred to as pullback, a convention we generally adopt. They are also sometimes called base-change functors. When and , we may write for the object if there is no chance of confusion. (Compare that the choice of is also suppressed in the notation for a pullback) We will not go into a comprehensive description of the theory of fibrations, but simply give a few basic results. We will give some examples in the next section. For a textbook treatment, see eg. Reference (chapters 1, 9), or Reference (chapter 8). Note that we will not give a formal definition of the term "pseudofunctor" here. See eg Reference , def. 1.4.4. Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-KUW7/ efr-KUW7 /efr-KUW7/ Definition Let be a pseudofunctor. Then there is a category defined as follows: The objects are pairs The morphisms are pairs Composition is given by the "chain rule" There is an obvious forgetful functor . The category is known as the Grothendieck construction of Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-TQ3W/ efr-TQ3W /efr-TQ3W/ Proposition is a Grothendieck fibration, and the pseudofunctor it induces is equivalent to Eigil Fjeldgren Rischel2025428Proof Given and it is clear that the map given by is locally Cartesian---the required bijection is the definition of maps in the Grothendieck construction. But it's straightforward to see that these compose. Given a pseudofunctor it is obvious that the assignment is pseudofunctorial as well (the required natural isomorphisms are just the formal opposites of the ones for ). Applying this through the equivalence of fibrations and pseudofunctors leads to the fiberwise opposite of a fibration. Explicitly: Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-8LGE/ efr-8LGE /efr-8LGE/ Corollary Let be a fibration. Then there exists a category , called the fiberwise opposite of , whose objects are the same as , and where a morphism is a tuple . Eigil Fjeldgren Rischel 2025 4 29 https://erischel.com/efr-ZCTD/ efr-ZCTD /efr-ZCTD/ Optics and Lenses We have already given somewhat of an account of lenses, optics, and their applications in the introduction. We briefly review the theory here, especially to normalize the notation and definitions. The best general source for this material is still Reference . The definition of optic relies on the notion of coend, which we briefly recall, see Reference for a textbook account. If is a functor, the coend is defined as the initial object receiving a map for each , so that for each , the square commutes. If has enough colimits, we may express the coend as the coequalizer of the diagram where the two maps are given on each component by and , respectively. (This is Remark 1.2.4 in Reference ). Note that in particular this implies that if has all colimits, then it also has all coends. Eigil Fjeldgren Rischel 2025 4 29 https://erischel.com/efr-MMQZ/ efr-MMQZ /efr-MMQZ/ Optic Definition Let be a monoidal category which acts on two categories . Then the category of optics has Objects pairs The set of morphisms given by the coend Given two optics with representatives their composite is given by where we omit coherence morphisms. When is a monoidal category acting on itself by tensor, we write Note that if are symmetric monoidal and these actions are symmetric, inherits a symmetric monoidal structure given by . (Also given a braiding one can induce a non-symmetric monoidal structure, but this almost never comes up). Eigil Fjeldgren Rischel 2025 4 29 https://erischel.com/efr-CRYH/ efr-CRYH /efr-CRYH/ On notation Remark Objects and morphisms in the category of optics have two parts---one going "forwards", in the same direction as the optic, and one going "backwards". In Reference the objects are written where is the forwards part. Hedges' work on open games used the binomial notation but wrote the forwards part on top. To make the connection to fibrations more natural, we instead write the forwards part on the bottom, . It is the backwards part which depends on the forwards part, hence the forwards part is the base of the fibration (when one exists)---and every part of the language of fibrations is built around a mental model where the base is at the bottom and the fibers are over it (including the word "base"). When reading the references, this may cause some confusion, but hopefully this can be overcome. While we're at it, let us note that when talking about optics we will freely use terms like "the forwards part" "the backwards object" and so on---the meaning of this should now be clear. Of course, once we get to cooptics/charts, this would be more than a little confusing, since in that case both components are in the same direction. In those cases we will speak of either the primary (forwards) part or the secondary (backwards) part, or use the language of fibrations and speak of the map or object "in the base" and "in the fiber". As noted above, the coend consists of triples up to the equivalence relation which, for every identifies the two tuples and . Note that this relation is not assumed to be inherently an equivalence relation---one takes the transitive-symmetric closure as usual. We call this relation the sliding relation (because we slide the map from the backwards part to the forwards part). Eigil Fjeldgren Rischel 2025 8 25 https://erischel.com/efr-50T5/ efr-50T5 /efr-50T5/ Remark Suppose are small categories. Then the coend defining is a small colimit of small sets, hence again small. Since clearly the set of objects is small, is again a small category. However, if are merely assumed to be locally small, we can not guarantee the same is true of , since the hom-sets are now defined by a coend/colimit with large indexing category. However, in many special cases, it can still be seen to be locally small, such as in the Cartesian case (where is clearly locally small). In this thesis, we will not delve further into this subtlety, simply working inside some universe where all our categories are small. Of course, we can also let the arrows in go in the same direction as . This does not seem to have played any role in the literature, but we will give this a name, as it is a useful example to have in mind for Markov fibrations (where we will construct our dependent optics, conceptually, as a fiberwise opposite) Eigil Fjeldgren Rischel 2025 4 29 https://erischel.com/efr-G0YN/ efr-G0YN /efr-G0YN/ Co-Optics Definition Given acting on , the category of co-optics, has objects pair and morphisms given by the coend Eigil Fjeldgren Rischel 2025 5 7 https://erischel.com/efr-J0WW/ efr-J0WW /efr-J0WW/ Remark For any monoidal category , . One way to think of an optic is as a string diagram , but which has a hole with space for a morphism . One inserts such a morphism by composing the optic with the optic representing it. This idea of "open diagrams" has been developed in much more detail by Román, Reference . Eigil Fjeldgren Rischel 2025 5 7 https://erischel.com/efr-UNIN/ efr-UNIN /efr-UNIN/ Proposition Suppose is semicartesian---in other words, that is terminal. Then there is a functor , which takes a to , and pair to the composite . In the case of the map is a bijection. The fact that in states (maps from the monoidal unit) on are given by states on , while costates are given by maps plays an important role in the use of optics to describe open games. See § for more on this. Let us say a few things about lenses. Eigil Fjeldgren Rischel 2025 5 7 https://erischel.com/efr-M19V/ efr-M19V /efr-M19V/ Proposition If is Cartesian monoidal, Eigil Fjeldgren Rischel202557Proof Here we use a general fact about coends, that ---this has been called the ninja Yoneda lemma, see Reference . In this case we sometimes write for the category . We have the following fact: Eigil Fjeldgren Rischel 2025 5 7 https://erischel.com/efr-2793/ efr-2793 /efr-2793/ Proposition The functor , , is a fibration. The fiber has the following description: Its objects are the objects of . A map is a map . The composite of is given by composing the two into then using the diagonal. Given the pullback functor is given by precomposing by . Eigil Fjeldgren Rischel 2025 5 7 https://erischel.com/efr-VTPS/ efr-VTPS /efr-VTPS/ Proposition If moreover admits pullbacks, there is a fibred functor which carries an object to and a morphism to the map over . This is fully faithful. This is the first way to see the maps of as dependent lenses---they receive the category of lenses as a full subcategory. Eigil Fjeldgren Rischel 2025 5 7 https://erischel.com/efr-TZ0P/ efr-TZ0P /efr-TZ0P/ Remark The squares are pullbacks in any category with products, even if it does not admit pullbacks in general. It follows that the full subcategory of spanned by objects of this form is always a fibration over , which is isomorphic to ---the fiberwise dual is isomorphic to . Eigil Fjeldgren Rischel 2025 4 18 https://erischel.com/efr-E3HE/ efr-E3HE /efr-E3HE/ Markov categories Recall that Markov categories (Fritz, Reference ) are semicartesian symmetric monoidal categories equipped with a suitably coherent system of comonoids for each object . These have been used as the basis for the categorical analysis of probability theory for a number of years, see eg Reference , Reference , Reference , Reference . See Reference for the basic theory, and Reference for the theory of representable Markov categories. We assume familiarity with this theory in general, but we will review some theory below, as well as a few new minor results which are helpful for the theory of Markov fibrations. The basic idea of a Markov category is to interpret morphisms as "stochastic processes" or kernels---that is, functions valued in probability measures. A morphism is a parametrized joint probability measure---the comonoid structure allows us to build a canonical such given parametrized measures (by precomposing their tensor with the map). This is the product measure of the two---the fact that in general not every map has this form (because is not necessarily Cartesian) allows us to express the probabilistic dependence---as in, non-independence---of one variable on another. A morphism is called deterministic if it is a comonoid (co)homomorphism, which amounts to the claim that ---in other words, that running two independent copies of the kernel (with the same input) is equivalent to running one and copying the output. The deterministic morphisms form a Cartesian monoidal subcategory which is denoted . We first note the following alternative characterization of Markov categories in terms of their deterministic morphisms. Eigil Fjeldgren Rischel 2025 5 2 https://erischel.com/efr-5B0J/ efr-5B0J /efr-5B0J/ Premarkov structure Definition Let be a symmetric monoidal category. A premarkov structure on is a wide symmetric monoidal subcategory ---that is, a class of morphisms which contains all identities and structural isomorphisms, and is stable under composition and monoidal products---so that the monoidal category is Cartesian. Eigil Fjeldgren Rischel 2025 5 2 https://erischel.com/efr-10HQ/ efr-10HQ /efr-10HQ/ Proposition Let be a monoidal category. Given a premarkov structure , there is a unique Markov structure on so that each morphism in is deterministic. Given a Markov structure, is a premarkov structure. A premarkov structure has the form for some Markov structure if and only if it is maximal. Eigil Fjeldgren Rischel202552Proof The existence part first claim is clear: acquires a unique Markov structure since it is Cartesian, and the inclusion of that Markov structure into gives a Markov structure on . Conversely, suppose is a Markov category and is a class of deterministic morphisms which is still Cartesian. This means the projections still exhibit as a product in . But since the pairing are the unique map lifting two given maps the pairing must be preserved by the inclusion . Since the canonical Markov structure is given as a pairing, this means the markov structure induced by must agree with the one given by , which is just the original one. The second claim is straightforward---see eg Reference remark 10.13. Now suppose , where is some larger premarkov structure. By the argument above, they must generate the same Markov structure on . But again, this implies that every map in is deterministic for this Markov structure, so we have . This finishes the proof. Note that given a merely monoidal category, we can ask whether it is Cartesian, and if it is, it admits a unique symmetry induced by the universal property of the product. Hence if is a Cartesian wide monoidal subcategory, we may attempt to define a symmetry on simply using the one from . However, it is not automatic that this symmetry is natural for all the morphisms in . This basic idea was already noted by Fritz (and goes back to Golubtsov's work in Reference , an important part of the prehistory of Markov categories), although the precise statement above appears to be novel. Since the coherence conditions required of the comonoids in a Markov structure can be somewhat hard to remember, this characterization may be easier to understand. Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-38FR/ efr-38FR /efr-38FR/ Representable Markov category Definition A Markov category is representable if the inclusion admits a right adjoint. In this case we denote the right adjoint and call the object for a distribution object for . Observe that (by definition,) and hence where we denote the induced monad on by an abuse of notation. Eigil Fjeldgren Rischel 2025 4 27 https://erischel.com/efr-OSN1/ efr-OSN1 /efr-OSN1/ Commonly used Markov Categories Example Here are some important Markov categories: (Reference , section 4) is the category whose objects are measurable spaces, whose morphisms are Markov kernels, with composition given by the Chapman-Kolmogorov equation and monoidal structure given by product measures. (Reference , section 4) is the full subcategory of spanned by the standard Borel spaces, that is by those measurable spaces arising as the Borel -algebra on separable, complete metric space. is the subcategory of spanned by finite sets in the powerset -algebra. A morphism in is equivalently a matrix with entries in and with for each (this is what is called a stochastic matrix). Let be the monad which assigns to the set of countably-supported probability measures. Then the Kleisli category is a Markov category, sometimes called the category of discrete probability. (Reference , example A.1.4) is the category of Tychonoff topological spaces and kernels which are valued in Radon probability measures, and where the measure varies continuous in with respect to the weak topology---in other words, given any continuous function the resulting function on given by is continuous. (A space is Tychonoff if it is Hausdorff and, given closed and not in , there exists continuous with for . Every locally compact Hausdorff space is Tychonoff). Eigil Fjeldgren Rischel 2025 4 27 https://erischel.com/efr-NKD7/ efr-NKD7 /efr-NKD7/ Diagram Markov Categories Example If is a Markov category and is any ordinary category, there is a Markov category whose objects are functors and whose morphisms are natural transformations between these considered as functors into (i.e natural transformations with stochastic components). The monoidal and Markov structure is defined simply component-wise. In particular, taking the walking arrow, we obtain a Markov category of deterministic arrows . We will denote this category simply . Again, the objects of this category are the deterministic morphisms of , while the morphisms are the commutative squares with not-necessarily-deterministic sides Eigil Fjeldgren Rischel 2025 4 27 https://erischel.com/efr-4VOU/ efr-4VOU /efr-4VOU/ A note on terminology Remark When we speak of a stochastic map in a Markov category, we always mean a map which is not necessarily deterministic---that is, a general map of . Given a map into a tensor product, the composites with the projections are called the marginals. Given two maps , we refer to any map with those marginals as a pairing of the two. Note that there is always a canonical pairing given by . We write for this canonical pairing. For deterministic maps this is just the usual pairing using the universal property of the product. (We mostly use this in cases where one map is deterministic, so that the pairing is unique assuming positivity). When , we say , and say the two coordinates are independent given (or just independent). Recall that the codomain functor is a fibration if and only if admits pullbacks. Since Markov categories have terminal objects but not, in general, products, they clearly cannot be expected to have pullbacks. However, if is positive (Reference , def. 11.22---but see Proposition below), given deterministic, every has a unique pairing with . Computing pullbacks in the subcategory we usually have a similar property. This will be the basic idea behind Markov prefibrations. By the following straightforward proposition, this property of having unique deterministic pairings is in fact equivalent to positivity: Eigil Fjeldgren Rischel 2025 4 26 https://erischel.com/efr-OYB6/ efr-OYB6 /efr-OYB6/ Characterization of positivity Proposition Let be a Markov category. The following are equivalent: In , if has deterministic marginal then . In other words, anything is independent of a deterministic variable. Given a deterministic morphism and any morphism there is a unique with those marginals. is positive. Eigil Fjeldgren Rischel2025426Proof Clearly 1 and 2 are equivalent, since independence just means is the independent pairing of the marginals---if it is uniquely determined by its marginals, it must be equal to the independent pairing, and conversely if it is necessarily independent, it is determined by its marginals. Now let us show this is equivalent to positivity. First suppose has unique pairings in this sense. Let be as in the definition of positivity. Then the two maps indicated are pairings of and , and since is deterministic, they are identical by hypothesis. Finally, the implication is precisely Reference , prop. 12.14. Eigil Fjeldgren Rischel 2025 3 27 https://erischel.com/efr-T3HM/ efr-T3HM /efr-T3HM/ Pullback-positive Definition Let be a Markov category. We say is pullback-positive if admits pullbacks and, given a diagram of this form where is deterministic, there is a unique map making the squares commute. Note that a pullback-positive category is in particular positive by taking . Eigil Fjeldgren Rischel 2025 9 14 https://erischel.com/efr-A7K7/ efr-A7K7 /efr-A7K7/ Remark The pullbacks appearing in Definition are of course pullbacks in , not , analogously to how is a product in , not . We will still use the notation for these pullbacks, which should not lead to any confusion. We may occasionally write instead of , if we are carrying out a construction which primarily involves . Since essentially no Markov categories have Cartesian products (except when the tensor product is Cartesian), this should also not lead to any ambiguity. Eigil Fjeldgren Rischel 2025 3 27 https://erischel.com/efr-0YP1/ efr-0YP1 /efr-0YP1/ Lemma Suppose any category admits products and intersections---that is, pullbacks whenever are subobjects of . Then it admits all finite limits. Suppose admits and preserves pullbacks along monomorphisms. Suppose further is positive. Then it is pullback-positive. Eigil Fjeldgren Rischel2025327Proof To see the first point, let be arbitrary maps. Note that , where the horizontal map is given by and the top by ---in the sense that the universal property of this intersection is exactly the universal property of the given pullback. Now suppose preserves this intersection, and is positive. (Note that it doesn't follow that the inclusion preserves the pullback , because it doesn't preserve the products). This amounts to the claim that a map lifts to the pullback if and only if the composite lifts over the map (note that the pullback of a monomorphism is a monomorphism). This implies in particular such a lift is always unique. Since if is positive, given , if the latter is deterministic there is a unique pairing this implies there is at most one map pairing the two. On the other hand, since the composite is deterministic and equal to the composite , they are both deterministic, and hence the pairing factors over the diagonal. Applying positivity again, to the tensor product since the latter component is deterministic, this pairing is independent, and hence the map does indeed lift, finishing the proof. The idea here is that a distribution on a subobject defined by some condition is simply a distribution so that the condition is satisfied with probability . This is a natural condition which holds in many Markov categories. The assumption that is pullback-positive will play a key role in the development of the theory of Markov fibrations. Although the theory could possibly be developed without assuming the base category has deterministic pullbacks, positivity seems to be an essential part. Eigil Fjeldgren Rischel 2025 4 27 https://erischel.com/efr-MW29/ efr-MW29 /efr-MW29/ Lemma If is representable and positive, has limits, and the monad preserves intersections, then is pullback-positive. Eigil Fjeldgren Rischel2025427Proof This is clear, since if the monad preserves a given limit, so does the inclusion into the Kleisli category (for completely abstract reasons), hence by Lemma we are done. We will need the notion of support in a Markov category, introduced in Reference , for certain examples, so we briefly record the definition and a few of its properties here. Eigil Fjeldgren Rischel 2025 4 3 https://erischel.com/efr-AB57/ efr-AB57 /efr-AB57/ Support of a morphism Definition Let be two morphisms in a Markov category. We say is absolutely continuous with respect to and write if, whenever two maps are -almost surely equal, they are also -almost surely equal. Let be a morphism in a Markov category. The support of , if it exists, is an object which represents the functor of morphisms into which are absolutely continuous with respect to . Eigil Fjeldgren Rischel 2025 4 3 https://erischel.com/efr-8MYE/ efr-8MYE /efr-8MYE/ Proposition The support of is equipped with a canonical deterministic monomorphism , so that a map into is absolutely continuous with respect to if and only if it factors over the support. Two maps are -almost surely equal if and only if they are strictly equal on the support. Eigil Fjeldgren Rischel 2025 6 18 https://erischel.com/efr-NC0R/ efr-NC0R /efr-NC0R/ Double Categories Eigil Fjeldgren Rischel 2025 6 18 https://erischel.com/efr-RXP4/ efr-RXP4 /efr-RXP4/ Double Category Definition A (strict) double category is a category internal to the category of categories. Concretely, it consists of: A set of objects A collection of vertical morphisms forming a category with A collection of horizontal morphisms forming a category with An a collection of squares. Each square has a left and right boundary given by vertical morphisms , and top and bottom boundary given by horizontal morphisms so that and so on: The squares compose horizontally and vertically in the obvious way, each of which form a category (in particular, there are identity squares for each vertical and horizontal map). A double functor is a mapping on objects, vertical and horizontal morphisms, and squares, which preserves all the identities and composition. There is also a notion of pseudo double category, which weakens the horizontal composition to only be associative and unital up to a coherent system of squares. We will not go into the details here, see § for more on this. Eigil Fjeldgren Rischel 2025 6 18 https://erischel.com/efr-BEJ6/ efr-BEJ6 /efr-BEJ6/ Example There is a pseudo double category where the objects are categories, the vertical maps are functors, the horizontal maps are profunctors (functors sometimes called bimodules), and the squares are natural transformations. For any category with pullbacks , there is a pseudo double category with as the vertical category, spans as the horizontal morphisms, and commutative diagrams as the squares. There is a double category of sets, functions, and relations. For any Markov category , there is a double category with and . For any category at all, there is a double category with and commutative squares as the pullback squares. Eigil Fjeldgren Rischel 2025 6 18 https://erischel.com/efr-Y1S5/ efr-Y1S5 /efr-Y1S5/ Definition A double category is thin if, for each compatible square of vertical and horizontal morphisms, there is at most one square filling it. In other words, such a square either commutes or doesn't. Sometimes the two classes of morphism are instead called loose and tight, especially in cases where the composition of the loose class is not associative, or if the loose class is a superset of the tight class. Although the explicit description above is probably the best way to think about the data of a double category, on a technical level it is often useful to think in terms of internal categories. An internal category in a category is a pair of objects , maps (the domain, codomain and identity), and a map (the multiplication), satisfying the usual laws of a category. A double category in the above sense is then an internal category in the category of categories. In the terms of Definition , is , and is the category whose objects are horizontal arrows, and whose morphisms are squares (composed vertically). Obviously, we could just as easily have oriented things horizontally, but this is the convention usually adopted. Given a double category , there is another double category called the transpose of , which has the same objects but exchanges the horizontal and vertical morphisms. In many cases it is not clear which of and is the "correct" one to work with, and we may have to pass back and forth between them. We try to stick to the convention that the horizontal morphisms are the "loose" ones. To avoid confusion, we will also simply specify the classes of morphisms directly, speaking for example of "the lenses" or "the charts" when working with the double category (Definition ). The concept of double category goes back to Reference . See Reference , section 12.3 for a textbook treatment. They have been widely used in applied category theory. An early application is in Reference , which constructed a "structured" version of the double category of cospans. Here the philosophy is that the horizontal morphisms are the "systems" (in a general sense) which are being wired together by horizontal composition, while the vertical morphisms (and squares) are "structure-preserving maps between systems". This is similar to the our double category of "bisystems" (§ ). Recent work by Lambert and Patterson Reference applies double categories to categorical algebra, with many applications to systems modeling, see Reference (as well as to classical category theory). In the next section we'll see their application to categorical systems theory, and later see them applied to parametrized morphisms. Eigil Fjeldgren Rischel 2024 6 30 https://erischel.com/efr-0023/ efr-0023 /efr-0023/ Review of Categorical Systems Theory We will give a review of the framework called categorical systems theory, due to Myers (Reference ). Eigil Fjeldgren Rischel 2024 6 6 https://erischel.com/efr-000A/ efr-000A /efr-000A/ Theory of Dynamical Systems Definition A theory of dynamical systems is an indexed category equipped with a section of its Grothendieck construction . It is called monoidal if is a monoidal category, and is a lax monoidal functor. It is further called symmetric monoidal if is a symmetric monoidal category and is a symmetric lax monoidal functor. Note that this is equivalent to requiring to be a (symmetric) monoidal fibration in the sense of Reference (see also Reference ). Eigil Fjeldgren Rischel 2024 6 30 https://erischel.com/efr-0024/ efr-0024 /efr-0024/ Lenses and charts Definition Let be an indexed category. A chart is simply a morphism in the Grothendieck construction . A lens is a morphism in the fiberwise opposite . Eigil Fjeldgren Rischel 2024 6 30 https://erischel.com/efr-0025/ efr-0025 /efr-0025/ Arenas Definition Given an indexed category , the double category of arenas has Objects the objects of ---note that these are the same as the objects of Vertical morphisms the lenses Horizontal morphisms the charts The double category is thin. Given a square of this form we can first project it to a square in . If this commutes, we can pull the lenses and charts back to a square in . The above square of lenses and charts will be said to commute if both of these squares commute. We will write arenas either as , or, for brevity when there is no need to discuss the two levels separately, simply with a symbol . We will also write both and as the situation calls for---there should be no confusion resulting from this. Eigil Fjeldgren Rischel 2024 6 30 https://erischel.com/efr-0028/ efr-0028 /efr-0028/ Dynamical system with interface Definition Let be an arena. A dynamical system with interface is an object and a lens . A morphism of systems is a map so that this square commutes in : The category of systems with interface is denoted Eigil Fjeldgren Rischel 2024 6 30 https://erischel.com/efr-0029/ efr-0029 /efr-0029/ Proposition Given a lens postcomposition gives a functor . Given a chart , there is a profunctor where the set over and is the set of maps so that this square commutes: This defines a double functor . The fibration encodes what sort of information can be "indexed over a space", while the section tells us what sort of information (such as a next step, or a gradient vector) must be produced to give a dynamical system. In this respect, the theory is very similar to the coalgebraic approach to dynamical systems, or systems that "do something", and indeed we have the following: Eigil Fjeldgren Rischel 2024 6 30 https://erischel.com/efr-0026/ efr-0026 /efr-0026/ Example Let be a functor. Then there is a dynamical systems theory where , is constant at , and . It is easy to see that the category of closed dynamical systems in this theory is equivalently the category of -coalgebras. Of course, coalgebras can already encode systems with input and output---the point of the CST framework is to separate out the input and output of systems so that they can be acted on in a compositional manner. Eigil Fjeldgren Rischel 2024 7 2 https://erischel.com/efr-002F/ efr-002F /efr-002F/ The theory of discrete dynamical systems Example There is a theory of dynamical systems with category of spaces , category of bundles (with pullbacks for reindexing), and tangent bundle . We call this the theory of discrete dynamical systems. (In the sense that they are both discrete-time and discrete-space). In another direction, we have the following comparison result: Eigil Fjeldgren Rischel 2024 6 30 https://erischel.com/efr-0027/ efr-0027 /efr-0027/ Example In the theory of discrete dynamical systems: The category of lenses is equivalently the category of polynomial functors and natural transformations between them. Using this equivalence, the category of dynamical systems with interface is exactly the category of -coalgebras and homomorphisms The first part here (which works, suitably formulated, for any locally Cartesian closed category), is a classical part of the theory of polynomial functors, see eg Reference . The second part is due to Spivak, see Reference . There is an extensive body of work on the description of interacting (discrete, deterministic) dynamical systems in terms of polynomial functors, see e.g. also Reference . Eigil Fjeldgren Rischel 2025 5 7 https://erischel.com/efr-GXJ0/ efr-GXJ0 /efr-GXJ0/ The theory of stochastic discrete dynamical systems Example There is a systems theory where (with the evident reindexing maps) and . In this theory, a closed dynamical system is a set equipped with a map . An open dynamical system has stochastic update function, but output which depends deterministically on the current system state. In fact this example works for any monad on the category of sets. Eigil Fjeldgren Rischel 2025 6 17 https://erischel.com/efr-KZEM/ efr-KZEM /efr-KZEM/ The theory of smooth dynamical systems Example There is a dynamical systems theory where is the category of smooth manifolds, is the category of fiber bundles over , with given by pullback along (note that pullbacks of bundles always exist), and with being the tangent bundle of in the ordinary sense. In this case, closed dynamical systems are manifolds equipped with (smooth) vector fields, which is what is classically thought of as a smooth dynamical system. Eigil Fjeldgren Rischel 2025 5 7 https://erischel.com/efr-9WQU/ efr-9WQU /efr-9WQU/ Clock system Example Consider the theory of discrete-time, discrete dynamical systems from Example . Let be a system. Consider the system given by in the forwards direction, and in the backwards direction. (To be clear, the object denoted is the map ---it is a singleton in each fiber). Then a morphism of systems consists of the following data: A function . Another function . A third function so that . So that and That is, it is a choice of a sequence of points in the state-space, and a sequence of inputs compatible with the outputs, so that this sequence obeys the dynamics. In other words, it is a trajectory of the system. Because of this, one views a generic chart map as a generalized trajectory, of a type given by the domain system. As another example, taking the state space to be with an update map that carries to one finds a system which classifies -periodic trajectories. Similarly, in the smooth case, the system classifies solutions of a smooth differential equation (those which extend to infinity). Recent work by Lynch, Myers, Staton, and the author (Reference ) constructs clock systems for theories of stochastic, discrete-time doctrines, although we will not delve into this here. Because this thesis is about Markov fibrations, we have chosen to present these ideas in terms of fibrations equipped with sections. Myers' book Reference actually prefers the presentation in terms of indexed categories. Similarly, we construct a double category of systems which is fibred (in a certain sense) over the double category of arenas. Myers instead displays this as a doubly indexed category , which carries lenses to functors and charts to profunctors. In a recent paper Reference , Myers and Libkind further develop the category theory of what they term double operadic categorical systems theory, which again concerns notions of "composable system" which are described in terms of such doubly indexed category (although for technical reasons, they use the language of right modules in that paper and a somewhat different presentation, the concept is the same.) Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-GFG0/ efr-GFG0 /efr-GFG0/ Open Games with external choice in Markov Fibrations Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-7EHV/ efr-7EHV /efr-7EHV/ Introduction Game theory is a field of economics which studies mathematical models of human decisionmaking. Classically, game theory is particularly interested in the behavior resulting from individual agents optimizing simple objectives (such as the expected value of some real-valued function of each players' decision) when multiple such players interact. Although the analysis of games, obviously, has a long prehistory, the field was put on its modern theoretical footing by Von Neumann in 1928 (Reference , see also Reference for a more thorough treatment from this era). The basic problem studied here is this: given two finite sets of strategies---that is, the choices available to the two players---and a function which assigns to a pair of such choices a score, which player 1 seeks to maximize and player 2 seeks to minimize, what can be said about the rational decisionmaking of each player? Suppose each player has access to some source of randomness, so that they may each choose a distribution on their respective strategy set to play. This is called a mixed strategy. Suppose player 2 has the opportunity to move knowing the distribution chosen by player 1 (but not the value drawn from it), and suppose he prefers to minimize the expected outcome of the game. Writing for for brevity, if player 1 selects the distribution , clearly player 2 must select the distribution . Knowing this, player 1 will select the distribution and the expected score of the game will be If the selection happens in the other order, of course, the result will be Clearly choosing knowing your opponent's (mixed) strategy can not be a disadvantage compared to choosing with no information, and so we have Von Neumann's great minimax theorem is that these values agree, and this implies that by choosing as if your opponent would know your (mixed) strategy, you obtain a result as good as you could have obtained if you knew your opponents' strategy---hence neither player could improve their outcome, even if they had the advantage of greater information. This is a strong argument for the optimality of such decisions. The key assumption is that the players are perfectly opposed, that is, player 2 seeks to minimize precisely the value that player 1 seeks to maximize. Nash in Reference extended the theory to the more general class of games where players simply each have their own utility function, although it should be noted that Nash merely proved the existence of equilibria---that is, strategy sets where no player can improve their situation by switching strategies. It is computationally intractable (Reference ) to identify Nash equilibria in an arbitrary game, and so to apply the theory we're forced to analyze each game in an ad hoc way. We will not go intro a serious review of game theory here, we mention the above details mainly to contextualize our future definitions. For a modern reference on game theory see eg Reference , or Reference for a textbook treatment. In his thesis (Reference ), Hedges introduced a novel approach to game theory which he named compositional game theory, studying objects called open games. The idea of open games is to describe a type of "partial" game which has an interface to the world---some part of the payoff function being dependent on an undetermined environment, into which hole can be inserted another game. This is the sense in which they are open. Based on these, one can build a complex game up out of simpler subparts, as well as leverage a string diagramattical syntax to analyze games. The original definition of open game cannot help but seem somewhat ad hoc. It was quickly realized that a large part of the definition can be understood to say that a game is a function from a strategy set into a set of lenses, and this part of the game composes by lens composition. The additional data of an open game is a so-called equilibrium relation, which determines which strategies are equilibria in a given situation (each strategy really represents a set of strategies, one for each player). In Reference , the author, Capucci, Gavranovic and Hedges demonstrated that this can be further simplified by realizing an open game as a parametrized map in lenses---a morphism in (in fact, one often wants to consider for, for example, Markov categories , to account for mixed strategies) along with a notion of equilibrium relation defined on the parameter object . We will begin this chapter by recapping this idea. On a conceptual level, there is a natural operation on games, called external choice, which assigns to two open games a new game, whose environment begins by making a choice between one of these games, lets that one happen, then provides some payoff for it at the end. The natural type for the interface of this operation is but unfortunately the category of optics doesn't have coproducts. The category of lenses can be extended with coproducts (into dependent lenses), but the analysis of mixed strategies makes probability an absolute necessity for a useful theory of games. The introduction of Markov fibrations provides a solution to this problem, and in the second half of this chapter, we provide such an external choice operation on open games. Although the approach based on Markov fibrations is novel, the approach to defining the external choice operator is otherwise very similar to one appearing in presently unpublished work by the author, Braithwaite, Hedges and Videla (this paper used a more specialized approach to adding coproducts to ). Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-CA6T/ efr-CA6T /efr-CA6T/ Open games in Monoidal categories Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-EGK8/ efr-EGK8 /efr-EGK8/ Definition Let be a monoidal category. A selection relation on an object is a relation . We write for the statement . Selection relations are ordered by inclusion, and so form a (posetal) category, which we denote . Given we define the pushforward on selection relations by . In other words, if and only if there exists so that and . It is clear that pushforward is monotone, so that this defines a functor Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-YKR3/ efr-YKR3 /efr-YKR3/ Remark Of course, there is an equally good "pull-back" operation on selection relations given by (this is the pushforward in ). These are adjoint representatives of the same profunctor, which should arguably be regarded as the primary object of interest---that is, we could work with a functor , the category of categories and profunctors, where we say two selection relations are related by if . However, we will stick with the pushforward definition for now, since it is conceptually simpler and good enough for our purposes. Eigil Fjeldgren Rischel 2025 5 5 https://erischel.com/efr-BNB2/ efr-BNB2 /efr-BNB2/ Example Let . There is a selection function defined by if and only if is a maximum of . (Identifying maps with points and maps with functions ) In the same category, for each there is a selection function given by . This corresponds to satisficing at the value (Reference )---that is, selecting any strategy which achieves this value or greater. For with semiCartesian, there is a selection function (using the same identification as above). The agents with this selection function are called predicting agents in Reference . These agents attempt to predict the value the environment will return to them. For the fiberwise opposite of manifolds and smooth bundles---that is, the category of lenses between smooth manifolds---there is a selection function on the tangent bundle given by if and only if ---that is, if is a fixpoint of the dynamical system identified by . Let be a dynamical systems theory and work in the category of lenses. Suppose the monoidal structure on is Cartesian, so that a lens is the same as a map . Note that has a distinguished section given by the identity. Then there is a selection function on each object where a lens , given by is in equilibrium with respect to a lens if is a trajectory between those systems---that is, if it is an equilibrium state of the smooth dynamical system . This subsumes the two previous examples. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-PTAE/ efr-PTAE /efr-PTAE/ The Nash Product Definition Let be selection relations. Their Nash product is given by Let us unpack this very dense definition. Given a context , a strategy is in equilibrium if: It decomposes as a tensor product of . Composing with , we obtain a map . This is the context of the first player assuming the second player plays . must be an equilibrium strategy for this map. Simultaneously, must be an equilibrium for the analogous composite of and . Eigil Fjeldgren Rischel 2025 5 5 https://erischel.com/efr-5E1V/ efr-5E1V /efr-5E1V/ Example Consider the selection relation on . A point is an equilibrium for a function if and only if maximizes and maximizes . In other words, if is a Nash equilibrium (Reference ) in the usual sense for the game with payoff matrix . Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-Z2BR/ efr-Z2BR /efr-Z2BR/ Proposition The Nash product along with the map given by the full set equips with a lax monoidal structure. Eigil Fjeldgren Rischel202554Proof It is trivial to verify associativity and unitality of the monoidal structure. The only hard part is to verify that is actually a natural transformation. To that end, let and be morphisms of . Let . First consider the statement . This holds if and only if factors as so that . This in turn means and and analogously for . Now consider the statement . This means that factors as so that and analogously for , which in turn means that factors as so that . Using the axioms of a monoidal category, it is straightforward to see that these two requirements on are equivalent, hence we have our naturality. Eigil Fjeldgren Rischel 2025 5 5 https://erischel.com/efr-UI90/ efr-UI90 /efr-UI90/ Corollary Since is lax monoidal, its Grothendieck construction acquires a monoidal structure (Reference ). We write this category ---explicitly, it is given as follows: The objects are pairs where and is a selection relation on it The morphisms are morphisms so that, for every , The monoidal structure is given by Now we are ready to make the slick definition of open games. Eigil Fjeldgren Rischel 2025 5 5 https://erischel.com/efr-1WGM/ efr-1WGM /efr-1WGM/ Definition Let be a semiCartesian symmetric monoidal category. The symmetric monoidal double category of open games in is . Note that in this case, and . Thus a selection function decides, for each payoff function , which of the states are suitable equilibria. Before we proceed to the case of stochastic lenses, we will pause briefly to make a small modification to the preceding theory as presented in Reference . In a game with forwards play function there are two ways to talk about the player's "choice"---we may say that the player chooses a strategy which then has some effect. Or we may say that the choice is really the and the strategy is the "precommitment" of choosing what to do given each possible . Once we introduce randomness---working in , for example---we see that there are two distinct ways for a player to make a random choice: first, his strategy may be stochastic---that is, he is choosing a random strategy. Or the morphism may be stochastic---this means each strategy contains a specification of how to randomly choose given each possible . In the case where , the distinction is between taking and letting the play function be the identity, and taking and letting the play function be the sampling map which stochastically draws an element from a distribution. We take the view that the latter is the proper presentation of this game---this goes along with the terminology in the classical game theory literature, which would certainly regard a distribution on the set of possible moves as a (mixed) strategy. Having represented our games like this, we may restrict ourselves to considering deterministic maps as strategies. This also fixes the awkwardness in the definition of the Nash product, since now every strategy in decomposes uniquely as a pair of strategies. We quickly modify the preceding definitions to make sense of this. Note that we also modify the reparametrization maps to be deterministic (in the base). Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-6I8U/ efr-6I8U /efr-6I8U/ Open games in a stochastic module Definition Let be a symmetric monoidal stochastic module fibration over the Markov category . Recall that acquires a symmetric monoidal structure. Denote as usual . Note that this is stable under the monoidal product, and acts on via the inclusion. Then the category of open games in is the category . When is a Markov prefibration, we overload the notation by writing . We now introduce the notion of strategic equivalence, which identified two open games if they have the same equilibria for every costate which can actually occur as a result of pasting the game into some larger diagram. Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-K0Y8/ efr-K0Y8 /efr-K0Y8/ Context Definition Let be objects of a symmetric monoidal category . A context for is a tuple . We denote the set of contexts . Given a morphism and a context the costate will be called the induced costate. Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-I1K5/ efr-I1K5 /efr-I1K5/ Remark The preceding definition clearly works in a non-symmetric monoidal category as well. In this case, arguably, a context should be defined to consist of maps . However, the problem with this from our point of view is that it doesn't allow the definition of a costate on given a parametrized map (since one cannot commute the past the ), which is what we're interested in. It is also worth observing that contexts are essentially the same thing as optics . We have not specified an equivalence relation on contexts (we do not need it,) but it is easy to see that for any parametrized map, two contexts that are sliding equivalent give the same induced costate. Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-R46X/ efr-R46X /efr-R46X/ Strategic Equivalence Definition Let be a monoidal stochastic module fibration over a Markov category , and consider a 2-cell . We say this is a strategic equivalence if the underlying map in is an isomorphism, and for every context , the induced costate on has the same equilibria (under this isomorphism) as the composite costate on Note that a game up to strategic equivalence is determined by a relation between strategies and contexts. This brings us closer to Hedges' original definition of open game from Reference . The chief difference is that a game in our sense is prevented from "inspecting" the context for those which are not in the image of , in the sense that whether a given strategy is in equilibrium or not cannot depend on this (since we only see a certain costate on ). Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-HLIX/ efr-HLIX /efr-HLIX/ Proposition Strategic equivalence is compatible with composition and tensor in . Eigil Fjeldgren Rischel202556Proof Let be strategic equivalences. By 2-cell composition there is a map , which we must show is a strategic equivalence. Let be a context in . Now a pair of strategies for are in Nash equilibrium for the costate induced by this context if and only if is in equilibrium for the costate induced by the context where is the play function of , and the analogous condition holds for . But if are equivalences, this is clearly equivalent to asking that be in Nash equilibrium for the costate induced by on . This concludes the proof. Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-U99C/ efr-U99C /efr-U99C/ Definition We denote by the symmetric monoidal category of strategic equivalence classes of open games. Note that this category of open games retains a 2-categorical structure, given by deterministic maps which preserve equilibria in every context. However, we will leave a deeper investigation of this structure for future work. Eigil Fjeldgren Rischel 2025 4 30 https://erischel.com/efr-SREZ/ efr-SREZ /efr-SREZ/ Open Games with External Choice Let be an extensive Markov category. Let be a monoidal stochastic module fibration with Markov structure, which has coproducts which are preserved by the pullbacks. Recall that acquires two monoidal structures: one from dualizing the given monoidal structure on which we simply denote , and one from taking the coCartesian monoidal structure in the fiber (which is Cartesian after taking the fiberwise dual, of course), which we denote . Note that is a Markov category. For the rest of this section, fix like this. Eigil Fjeldgren Rischel 2025 5 5 https://erischel.com/efr-Y2ZN/ efr-Y2ZN /efr-Y2ZN/ Lemma Let be objects in , and let be a morphism in . Then there is a canonical map so that the underlying map is Eigil Fjeldgren Rischel202555Proof The first map in the factorization has a deterministic retract (deleting the component,) and using the coproduct-preservation, the coproduct over and pull back to the same object over . Composing the induced stochastic-Cartesian map and the Cartesian map gives the canonical map we wanted. With the interpretation that is the object indexed over this map simply selects one branch randomly and marginalizes to that coordinate in then includes the returned value into the coproduct. Eigil Fjeldgren Rischel 2025 4 30 https://erischel.com/efr-98DR/ efr-98DR /efr-98DR/ Definition When as above, acquires a monoidal structure which we call external choice, and write , given on objects by the coproduct in and on morphisms by the following formula: Given two open games their external choice is has parameter . The play map is given by The selection relation is given (up to equivalence) as follows: Given a context and a deterministic state , they are in equilibrium if factors over the canonical for some . The factorization being given by ,and being given by , we have The idea behind the external choice operator is that, for all the contexts which can actually occur as a result of pasting a game into a larger string diagram, the map has the given form---that is, the probability of landing in each of the two fibers does not depend on the chosen and the conditional distributions on the component of the fiber depend only on . Hence we need only concern ourselves with which states are equilibria for contexts of this form. The choice of the empty set of equilibria for other contexts is merely a convention. Eigil Fjeldgren Rischel 2025 4 30 https://erischel.com/efr-VB6A/ efr-VB6A /efr-VB6A/ Given a state in a Markov category with coproducts, we say a pair form a pair of conditionals if the copairing is a Bayesian inverse of the map . Given a state in , we say a pair of maps form a pair of conditionals if the underlying maps do. Eigil Fjeldgren Rischel 2025 5 5 https://erischel.com/efr-MLF1/ efr-MLF1 /efr-MLF1/ Theorem As defined above, is a symmetric monoidal structure on . Eigil Fjeldgren Rischel202555Proof The monoidal coherences come from the monoidal structure of and it is trivial to see that they preserve the selection relations. The only nontrivial part is proving that is functorial. Hence let be games. The strategy set of is given by For by . In the base, these are the same object . In the fiber, they are given respectively by and Note the coproducts here are the fiberwise ones. There is an obvious lens from the former to the latter (given by the identity map on the base, and the inclusion of two summands in a fourfold coproduct---note that lenses go backwards in the fiber). Letting be a context, and going through the definitions, it is clear that the resulting contexts for the former game factors as this lens followed by the context for the latter game. In other words, this lens is a reparametrization map between the two games. It suffices to verify it is an equivalence. Unpacking the equivalence relation on , note that (in all the possible contexts,) the signal to does not depend on the action of and vice versa, and so for and . Hence they are in equilibrium if and only if they are in equilibrium in for the given context (conditioned on that branch), and similarly the other two. This proves the desired equivalence. Eigil Fjeldgren Rischel 2025 4 30 https://erischel.com/efr-4CSL/ efr-4CSL /efr-4CSL/ Example Consider the (Grothendieck) fibration , which can be viewed as a Markov fibration. Let be a parameterized lens. Denote by the open game with parameters , underlying parameterized lens , and equilibrium relation given by . Then if is another parameterized lens, we have where by an abuse of notation denotes the paramterized lens given by distributing into the coproduct, then projecting into the relevant factor of the product and applying either or A context for either of these games consists of an element of and a function . The external choice game can be seen as having two players, one who gets to play if the context chooses an element in , who must output an element and optimize (according to his private utility function ), the other playing when the input is in and who must choose an element in . The single argmax game can be seen as a single player, who is constrained to play inside the same "branch" of the game as the input (this constraint is encoded in the lens ), and whose utility function is given by the first players' in the first branch, and the second players' in the second branch.x Eigil Fjeldgren Rischel 2025 5 5 https://erischel.com/efr-BNT1/ efr-BNT1 /efr-BNT1/ Example Let be a game, representing an agent who is optimizing the return value in some sense. Then represents the same agent, whose payoff may now depend on an additional bit (a value in ), but whose decisions may not depend on that bit (his selection function may still depend on its distribution). On the other hand, represents the same situation, but where the player's strategy may depend on the bit---he provides two strategies , one for each possibility. This proves that does not distribute over Eigil Fjeldgren Rischel 2025 4 29 https://erischel.com/efr-ZCTD/ efr-ZCTD /efr-ZCTD/ Optics and Lenses We have already given somewhat of an account of lenses, optics, and their applications in the introduction. We briefly review the theory here, especially to normalize the notation and definitions. The best general source for this material is still Reference . The definition of optic relies on the notion of coend, which we briefly recall, see Reference for a textbook account. If is a functor, the coend is defined as the initial object receiving a map for each , so that for each , the square commutes. If has enough colimits, we may express the coend as the coequalizer of the diagram where the two maps are given on each component by and , respectively. (This is Remark 1.2.4 in Reference ). Note that in particular this implies that if has all colimits, then it also has all coends. Eigil Fjeldgren Rischel 2025 4 29 https://erischel.com/efr-MMQZ/ efr-MMQZ /efr-MMQZ/ Optic Definition Let be a monoidal category which acts on two categories . Then the category of optics has Objects pairs The set of morphisms given by the coend Given two optics with representatives their composite is given by where we omit coherence morphisms. When is a monoidal category acting on itself by tensor, we write Note that if are symmetric monoidal and these actions are symmetric, inherits a symmetric monoidal structure given by . (Also given a braiding one can induce a non-symmetric monoidal structure, but this almost never comes up). Eigil Fjeldgren Rischel 2025 4 29 https://erischel.com/efr-CRYH/ efr-CRYH /efr-CRYH/ On notation Remark Objects and morphisms in the category of optics have two parts---one going "forwards", in the same direction as the optic, and one going "backwards". In Reference the objects are written where is the forwards part. Hedges' work on open games used the binomial notation but wrote the forwards part on top. To make the connection to fibrations more natural, we instead write the forwards part on the bottom, . It is the backwards part which depends on the forwards part, hence the forwards part is the base of the fibration (when one exists)---and every part of the language of fibrations is built around a mental model where the base is at the bottom and the fibers are over it (including the word "base"). When reading the references, this may cause some confusion, but hopefully this can be overcome. While we're at it, let us note that when talking about optics we will freely use terms like "the forwards part" "the backwards object" and so on---the meaning of this should now be clear. Of course, once we get to cooptics/charts, this would be more than a little confusing, since in that case both components are in the same direction. In those cases we will speak of either the primary (forwards) part or the secondary (backwards) part, or use the language of fibrations and speak of the map or object "in the base" and "in the fiber". As noted above, the coend consists of triples up to the equivalence relation which, for every identifies the two tuples and . Note that this relation is not assumed to be inherently an equivalence relation---one takes the transitive-symmetric closure as usual. We call this relation the sliding relation (because we slide the map from the backwards part to the forwards part). Eigil Fjeldgren Rischel 2025 8 25 https://erischel.com/efr-50T5/ efr-50T5 /efr-50T5/ Remark Suppose are small categories. Then the coend defining is a small colimit of small sets, hence again small. Since clearly the set of objects is small, is again a small category. However, if are merely assumed to be locally small, we can not guarantee the same is true of , since the hom-sets are now defined by a coend/colimit with large indexing category. However, in many special cases, it can still be seen to be locally small, such as in the Cartesian case (where is clearly locally small). In this thesis, we will not delve further into this subtlety, simply working inside some universe where all our categories are small. Of course, we can also let the arrows in go in the same direction as . This does not seem to have played any role in the literature, but we will give this a name, as it is a useful example to have in mind for Markov fibrations (where we will construct our dependent optics, conceptually, as a fiberwise opposite) Eigil Fjeldgren Rischel 2025 4 29 https://erischel.com/efr-G0YN/ efr-G0YN /efr-G0YN/ Co-Optics Definition Given acting on , the category of co-optics, has objects pair and morphisms given by the coend Eigil Fjeldgren Rischel 2025 5 7 https://erischel.com/efr-J0WW/ efr-J0WW /efr-J0WW/ Remark For any monoidal category , . One way to think of an optic is as a string diagram , but which has a hole with space for a morphism . One inserts such a morphism by composing the optic with the optic representing it. This idea of "open diagrams" has been developed in much more detail by Román, Reference . Eigil Fjeldgren Rischel 2025 5 7 https://erischel.com/efr-UNIN/ efr-UNIN /efr-UNIN/ Proposition Suppose is semicartesian---in other words, that is terminal. Then there is a functor , which takes a to , and pair to the composite . In the case of the map is a bijection. The fact that in states (maps from the monoidal unit) on are given by states on , while costates are given by maps plays an important role in the use of optics to describe open games. See § for more on this. Let us say a few things about lenses. Eigil Fjeldgren Rischel 2025 5 7 https://erischel.com/efr-M19V/ efr-M19V /efr-M19V/ Proposition If is Cartesian monoidal, Eigil Fjeldgren Rischel202557Proof Here we use a general fact about coends, that ---this has been called the ninja Yoneda lemma, see Reference . In this case we sometimes write for the category . We have the following fact: Eigil Fjeldgren Rischel 2025 5 7 https://erischel.com/efr-2793/ efr-2793 /efr-2793/ Proposition The functor , , is a fibration. The fiber has the following description: Its objects are the objects of . A map is a map . The composite of is given by composing the two into then using the diagonal. Given the pullback functor is given by precomposing by . Eigil Fjeldgren Rischel 2025 5 7 https://erischel.com/efr-VTPS/ efr-VTPS /efr-VTPS/ Proposition If moreover admits pullbacks, there is a fibred functor which carries an object to and a morphism to the map over . This is fully faithful. This is the first way to see the maps of as dependent lenses---they receive the category of lenses as a full subcategory. Eigil Fjeldgren Rischel 2025 5 7 https://erischel.com/efr-TZ0P/ efr-TZ0P /efr-TZ0P/ Remark The squares are pullbacks in any category with products, even if it does not admit pullbacks in general. It follows that the full subcategory of spanned by objects of this form is always a fibration over , which is isomorphic to ---the fiberwise dual is isomorphic to . Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-K6NM/ efr-K6NM /efr-K6NM/ Theorem Let be any pullback-positive Markov category. Then is a Markov prefibration which thus induces a stochastic module structure on . Writing simply for , we have: There is a functor which is fully faithful. Dually there is a functor which is fully faithful. and both admit symmetric monoidal structures, which make the functors strict symmetric monoidal, as well as the functors strong symmetric monoidal. If is extensive, this functor preserves the coproducts and both admit all finite coproducts. If moreover has conditionals and supports, Eigil Fjeldgren Rischel 2025 4 8 https://erischel.com/efr-ZRUZ/ efr-ZRUZ /efr-ZRUZ/ Categories of stochastic dynamical systems Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-5QC5/ efr-5QC5 /efr-5QC5/ Introduction Given a set of inputs and a set of outputs , there are essentially two make sense of the informal description "finite-state automaton which reads inputs from and produces outputs in ". These are the notions of Mealy machine and Moore machine. Simply put, if the set of states is , a mealy machine is a function whereas a Moore machine is a pair . In other words, in a Moore machine the output does not depend on the current input, but only on previous inputs (through their effect on the state), but in a Mealy machine, the input can be passed through directly. Using the language of categorical systems theory, we can make the following definitions: Eigil Fjeldgren Rischel 2024 5 25 https://erischel.com/efr-0006/ efr-0006 /efr-0006/ Moore Machine Definition In a dynamical systems theory, a Moore machine with state space and interface is a lens . The category of Moore machines with interface is the slice category of the functor over ---that is, a morphism of Moore machines is a morphism so that the obvious triangle commutes. Eigil Fjeldgren Rischel 2024 5 25 https://erischel.com/efr-0005/ efr-0005 /efr-0005/ Mealy Machine Definition In a dynamical systems theory, a Mealy machine with state space and interface consists of a costate lens . The category of Mealy machines with interface is the comma category of the functor over ---that is, a morphism of Mealy machines is so that the obvious triangle commutes. A Moore machine in the sense of Definition is what Myers calls a (open) dynamical system, and they are the central object of study in Reference . Arguably, both Moore machines and Mealy machines deserve the name of "open dynamical system"---the difference is how they interact with the external world. Observe in particular that, if , the categories of Mealy and Moore machines agree, both being equal to the slice of over the unit . In other words, the two notions of closed dynamical system coincide. It is clear that both Mealy and Moore machines, in this sense, are special kinds of parametrized morphism in lenses, namely those parametrized by an object of the form . This leads naturally to the idea that there should be a triple category of morphisms of this type, charts, and lenses. We call the parametrized lenses bisystems since they generalize the two types of machine, Moore and Mealy (but we choose to stick with "system" rather than "machine"). Eigil Fjeldgren Rischel 2024 6 13 https://erischel.com/efr-000E/ efr-000E /efr-000E/ Machine Learning as a bisystem Example It is an observation which goes back at least to Reference , and was more thoroughly developed in Reference , that the process of training a machine learning algorithm by gradient descent can be abstracted in the following way: Wanting to learn a function , we choose a parametrized function , where all these are, in the simplest case, Euclidean spaces We take the backwards derivative of obtaining a lens: For each datum we compute the loss gradient , and combining this with and the current parameter , we get a gradient on the parameter space which we can use to update Thus a machine learning algorithm is a sort of bisystem. Indeed our bisystems are essentially an abstracted version of the learners of Fong--Spivak--Tuyeras. The functoriality of this assignment is the main point of the above-mentioned papers. In this chapter, we will put together the ingredients we have assembled so far and construct a triple category of stochastic dynamical systems. We will also give the construction of triple categories of systems in the ordinary case. Note that the theory of Markov fibrations does not quite generalize the ordinary theory of fibrations---only fibrations with a Cartesian base (Example and Proposition ). Since the pullbacks in play such a key role in the theory, it is not clear that this can be dispensed with. Although Cartesian bases certainly cover the vast majority of examples from the literature on categorical systems, it is of course worth noting that they are not a requirement. Moreover, we will see that the construction of the double category of lenses and charts encounters certain problems for a general Markov fibration not seen for ordinary fibrations. Hence we will give a separate description of the triple categories in each of the two cases. The notions of Mealy and Moore machine are both quite old, going back to Reference , Reference . While we do obtain finite-state automata of these types as special cases, our interest is primarily in the analysis of dynamical systems, which tends to ask rather different questions than automata theory. Thus, despite using the terminology, we will not be particularly interested in the actual theory of Mealy and Moore machines. We do mention one recent paper, Reference , which has a category-theoretic approach similar in spirit to our own. Their category of Mealy machines can be obtained, not as our category of Mealy machines above, but by considering maps in the case where are trivial in the secondary component. In the discrete case, such a map is given by . It should also be noted that the idea of embedding Mealy machines as the morphisms is not original, but was communicated to the author by Matteo Capucci. It seems not to have appeared in the literature so far. The idea that there "should" be a triple category of systems, like the one we will construct, has also circulated as folklore, although again an explicit construction has yet to appear. Very similar ideas appear in the work of Shapiro and Spivak, see for example Reference . We begin this chapter with a treatment of double categories of charts and lenses in the context of Markov fibrations (and stochastic modules). The construction does not work quite as well as in the classical case---the difficulty is essentially that the equivalence relation which defines stochastic charts has a directed nature, and given a 2-cell defined in an obvious way and an equation there is not (apparently) in general a way to lift this to an arrow (with ). However, we can construct a double category whose globular horizontal 2-category have connected components given by the stochastic charts (or lenses). We proceed to give an account of categorical systems theory for these double categories. The chief problem posed by the above is that we may not have any good clock systems (Example ). Two equivalent charts should represent the same system, but they may receive different sets of maps from the supposed clock system (and thus have different sets of trajectories). We resolve this by proving that, for lenses with a deterministic base, every equivalence class of lenses has an initial representative. Moreover, mapping out of this initial representative to a representative of some other lens, a 2-cell exists filling a given square if and only if it commutes as a map of lenses and charts in the classical case (recall that over deterministic bases, Markov fibrations become ordinary fibrations). In particular the trajectories of a system with respect to such a clock system depend only on their equivalence class. We follow this up by constructing the above discussed triple categories of "bimachines", that is systems which combine Mealy and Moore machines. We do this both for stochastic and the ordinary case---the procedure is exactly the same, but of course they are different objects, neither generalizing the other. We end by constructing a stochastic dynamical systems theory for smooth dynamical systems---requiring a brief detour to construct a suitable Markov category of smooth kernels. Eigil Fjeldgren Rischel 2025 4 24 https://erischel.com/efr-SIE7/ efr-SIE7 /efr-SIE7/ Double categories of stochastic charts and lenses To construct the double category of charts and lenses for an ordinary fibration , one can use the following procedure: Form the square double category Take the fiberwise opposite of these objects: . Observe that fiberwise opposite preserves pullbacks, and hence this is again a double category. Here we used the following result: If is a Grothendieck fibration and is any category, then is again a fibration (which classifies the lax limits of the composite ), and a natural transformation is Cartesian iff it is levelwise Cartesian. It would be neat to obtain a similar result for Markov fibrations. The first problem with this is that does not generally inherit a Markov structure from . As we noted when we introduced diagram Markov categories, one has to consider the category of deterministic diagrams instead. First, we will see that this indeed works for Markov prefibrations. This implies that lifts from fibrations to stochastic modules. Eigil Fjeldgren Rischel 2025 4 8 https://erischel.com/efr-6G7N/ efr-6G7N /efr-6G7N/ Proposition Let be a Markov prefibration. Recall that by we denote the category of deterministic arrows in . Let denote the ordinary arrow category. Let now denote the category consisting of those arrows in which lie over a deterministic base (but again, where the morphisms consist of commutative squares whose other sides do not necessarily have deterministic bases). Then is a Markov prefibration. This yields a functor , so that . This equation induces a natural transformation , which in turns gives a lift of to the category of stochastic module fibrations, where the induced algebra structure acts pointwise. Eigil Fjeldgren Rischel202548Proof Noting that is a Markov category with the "pointwise" structure, and the deterministic maps consist precisely of the pointwise deterministic maps, clearly , and fibrations are stable under the formation of arrow categories, with Cartesian maps formed pointwise. It is not trivial that this is a prefibration, because given a triangle in ---a "prism"---and a Cartesian lifting, we only know that the maps "at the ends" are Cartesian, not the maps between the triangles. Therefore we can not immediately apply the unique lifting property to say that the square between the induced lifts commute, given some map over . However, by taking the pullback on both sides (and noting that pullbacks are functorial,) we can factor this square into two which live entirely over a deterministic base, and where the Cartesian property therefore imply commutativity. We have already argued that this commutes with restriction to the deterministic part. The natural transformation is induced for completely abstract reasons, by applying to the unit to obtain a map which by the universal property of induces the desired map . Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-DXJR/ efr-DXJR /efr-DXJR/ Corollary Let be a small category, and let be a Markov prefibration. Let . Then the functor is a Markov prefibration. Eigil Fjeldgren Rischel202556Proof Note that is a limit of the categories and prefibrations are stable under these limits. The question is now If is a Markov fibration, we get a stochastic module structure on ---does it present a markov fibration? There is an induced map (where the latter is taken by convention to mean the full subcategory of functors whose image in consists of deterministic arrows). Is this an isomorphism? (If it is, clearly this implies point 1) Unfortunately it's not clear that either of these are true---the surjectivity of cannot a priori be lifted to the arrow category. The issue is that, given a map in consisting of, say it is not sufficient to find charts representing each of these---we must find a chart of squares representing the square. This is not guaranteed by the Markov fibration structure, and a similar issue comes into play for the equivalence witnesses. We may attempt to ignore this issue and simply try to form a double category given a stochastic module , but here the problem is that does not commute with limits in general. Hence we can not easily define a composition on the 2-cells of lenses obtained this way. There are various ways we might attempt to remedy this problem. One approach would be to formulate a behavioural notion of "commutativity" for squares of stochastic lenses and charts, but the problem with this is that it is not obvious whether this property is stable under composition. The basic problem stems from the fact that chart equivalences have a "directed" nature, and given a morphism of precharts and an equivalence (for example given by a stochastic section satisfying suitable conditions), there is not in general a way to lift this back into an equivalent with a map to . This observation leads to the idea that we might define a double category of precharts which has the directed equivalences among its morphisms (going only in one direction). We will begin by constructing this double category. Eigil Fjeldgren Rischel 2025 4 23 https://erischel.com/efr-KO4T/ efr-KO4T /efr-KO4T/ Definition Let be a (Grothendieck) fibration, and let be a faithful, identity-on-objects functor. Suppose admits pullbacks, and given a pair of morphisms over , where is in , there is a unique common factorization . The double category has as the vertical category, spans in equipped with a section as horizontal cells, and maps of spans which commute with the sections as 2-cells. The double category lying over has as the vertical category, and spans where is Cartesian, decorated with a section in as the horizontal cells, and maps of such spans (so that the underlying thing commutes with the sections) as the 2-cells. We will call the horizontal cells decorated spans. There is an apparent forgetful functor To spell it out, a 2-cell in consists of a diagram of this form in , where are Cartesian, and (writing for the object underlying , and so on) two sections of the underlying maps in , so that the second diagram also commutes in : Naturally, we are interested in the case of for a stochastic module . Then the decorated spans are representatives for stochastic charts. We will start by introducing a loosed notion of 2-cell for these spans, which combines the directed equivalences with the ordinary deterministic 2-cells of spans. Eigil Fjeldgren Rischel 2025 4 24 https://erischel.com/efr-RD28/ efr-RD28 /efr-RD28/ Definition Let be a Markov category and let be a stochastic module over . Let be decorated spans in , and let be morphisms in , so that we have a square A 2-cell of decorated spans for this data consists of a morphism (that is, possibly stochastic), satisfying the following two conditions. First, the diagram in the base must commute. Given this, there is an induced square in , where the bottom map is given by pulling back along , in the sense of Lemma . The second condition is that this square must also commute. Eigil Fjeldgren Rischel 2025 4 24 https://erischel.com/efr-DRU6/ efr-DRU6 /efr-DRU6/ Lemma 2-cells of decorated spans compose---that is, given a diagram where the horizontal maps are decorated spans, and the vertical maps are maps in with deterministic base, and given maps of decorated spans there is a map of decorated spans Eigil Fjeldgren Rischel2025424Proof We can easily compose the two maps to get . Now, the question is whether the perimeter of this diagram commutes: Note that the top square is commutative by assumption, since pullbacks compose (even along stochastic maps) and this is the assumption that is a cell. The bottom square is the result of pulling back a commutative square over again along . Note that pullback along stochastic morphisms is not in general functorial---but since the vertical parts of this square are themselves pulled back from , this composition is preserved by pullback along . This finishes the proof. Eigil Fjeldgren Rischel 2025 4 24 https://erischel.com/efr-L7X0/ efr-L7X0 /efr-L7X0/ Proposition 2-cells of decorated spans compose horizontally: Given a square where the horizontal maps are precharts and the vertical maps are morphisms in , and given and , there is a prechart morphism . Eigil Fjeldgren Rischel2025424Proof Unlike the proof of Lemma , this is straightforward: If the carrier of is , and of (for ), then by definition the composites are carried by the pullback . There is a canonical map over , given by the independent pairing of and . Since pullbacks compose, the square over that must commute is given by the two commutative squares induced by , pulled back and composed with each other. Here we are pulling back along the deterministic projections from the pullback, and hence these commutative squares are preserved, and hence the composite square commutes as well. Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-MUNG/ efr-MUNG /efr-MUNG/ Corollary Let be a stochastic module over . There is a double category which has decorated spans as its horizontal maps, morphisms in as its vertical maps, and decorated span 2-cells as its 2-cells. Moreover, there is another double category which has decorated spans in as its horizontal maps instead. (The modification of everything above to lenses instead of charts is obvious). In fact, the globular 2-cells are in a sense exactly the equations defining the set of charts: Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-CU3J/ efr-CU3J /efr-CU3J/ Lemma Let be parallel decorated spans in (or ). They represent the same stochastic chart (lens) if and only if there exists a zig-zag of globular 2-cells between them. Eigil Fjeldgren Rischel202556Proof Unwinding the definitions, a 2-cell is precisely a map exhibiting the equality of the two decorated spans, as in Lemma The following straightforward lemma shows that charts over deterministic bases have initial representatives as spans. This will be important later: Eigil Fjeldgren Rischel 2025 4 24 https://erischel.com/efr-10CM/ efr-10CM /efr-10CM/ Proposition Suppose given a square in (or ) Suppose further the underlying square in is deterministic. Then: If there exists a filling decorated span 2-cell, the image in commutes. If is the carrier of and the left leg is an isomorphism, then this implication is an equivalence. Eigil Fjeldgren Rischel2025424Proof Given a stochastic chart carried by so that is deterministic, recall that we can first pull back to the equalizer of the two maps , then along the prescribed section . Note that this gives a 2-cell from this canonical representative with carrier to the initial representative. Given two such cells, we get a square But as part of the square surrounding the 2-cell, there is given a map , which must make this square commute. The functoriality of base change (pulling back the map ) proves this bottom map is again a 2-cell. But the property for this map to be a 2-cell is exactly the property for this square to be a commutative square in . This proves both parts of the statement. Eigil Fjeldgren Rischel 2025 4 8 https://erischel.com/efr-7DV5/ efr-7DV5 /efr-7DV5/ Double Categories of Stochastic Dynamical System Eigil Fjeldgren Rischel 2025 4 25 https://erischel.com/efr-HS7G/ efr-HS7G /efr-HS7G/ Stochastic Dynamical Systems Theories Definition A stochastic dynamical systems theory consists of a stochastic module over equipped with a section of the underlying fibration. We adopt the notation for the double category which was denoted above. We use the symbol for the horizontal (span) morphisms and for the vertical morphisms (in ). Given a stochastic dynamical systems theory , the double category of dynamical systems has Objects triples Horizontal morphisms given by a pair where is a square filling Vertical morphisms given by a pair where is a square filling Eigil Fjeldgren Rischel 2025 4 25 https://erischel.com/efr-N5V5/ efr-N5V5 /efr-N5V5/ Proposition Let be a stochastic dynamical systems theory. Then is an ordinary dynamical systems theory. There is a double functor . This functor is full on vertical morphisms, and on 2-cells. Let denote the subcategory spanned by systems with deterministic readout, prelenses with deterministic base, and all the morphisms of . Then the restricted functor admits a chartwise right adjoint, which assigns to each system or prelens its equivalence class. Eigil Fjeldgren Rischel2025425Proof The functor simply acts as the functor in Proposition . Since that inclusion is full on 2-cells, this one is full on vertical morphisms, and since a 2-cell in is merely a 2-cell in between specific objects, it is also full on 2-cells. The right adjoint property likewise follows from the analogous property of the inclusion functor on arenas. Eigil Fjeldgren Rischel 2025 4 25 https://erischel.com/efr-JQI9/ efr-JQI9 /efr-JQI9/ Proposition Let be a Markov prefibration. Then there is a double functor which acts as identity on the morphisms of , and carries each lens to the prelens . This double functor is full on 2-cells. The restriction to admits a right adjoint, which carries every prelens to its equivalence class. Eigil Fjeldgren Rischel2025425Proof First we must verify functoriality with respect to lens composition. To that end, let be deterministic morphisms and be lenses over them. Their composite as prelenses has carrier and as the morphism in the fiber, which is exactly the prelens associated to their composite as lenses. Second, we must verify fullness on 2-cells. But this is a special case of Proposition . Finally, Proposition is precisely the statement that assigning a prelens to its equivalence class lens is right adjoint to this (and in particular that it forms a functor) Recall that in Myers' categorical dynamical systems theory, trajectories of a system are identified with chart morphisms from a "clock" system---thus for example trajectories of a smooth dynamical system are exactly those maps which, when is equipped with the vectorfield , are homomorphisms. In general this presents an issue for our replacement category ---since we wish to regard two systems given by equivalent lenses as equivalent, but their set of homomorphisms from a given clock system is not necessarily in bijection. In the general case, we do not presently have a way around this problem---but at least for clock systems with deterministic readout, the above presents a solution: choosing the initial representative for such a system, we find that the set of trajectories does not depend on the equivalence class of the target system. Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-MWCE/ efr-MWCE /efr-MWCE/ The triple category of dynamical systems Having done the prep-work in § (and § ), we can jump directly into the construction: Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-EPI9/ efr-EPI9 /efr-EPI9/ Definition Let be a symmetric monoidal dynamical systems theory. Then this diagram: depicts two strict double categories, each with a symmetric monoidal structure, and a strict double functor between them which is (non-strictly) a symmetric monoidal functor. This induces an object of . Applying under we obtain a symmetric pseudomonoid in internal pseudocategories in . Denote by this induced object. Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-WX1V/ efr-WX1V /efr-WX1V/ The Structure of Remark has the following structure: The objects are simply the objects of ---the bundles. There are three types of 1-cell: Lenses, that is morphism in , which we write , charts, that is morphisms in , which we write , and bisystems, which are pairs , and which we write . Moreover there are three types of 2-cell: Lens-chart cells, which are the 2-cells of . Note that this double category is thin. We will say a square of lenses and charts commutes if it is filled by such a 2-cell. Chart-bisystem cells---given charts and systems a square filling this is a choice of map so that the resulting lens-chart square commutes. Lens-bisystem cells. Given lenses and systems a 2-cell consists of an isomorphism so that the resulting square of lenses commutes (recall that isomorphism charts are the same as isomorphism lenses). The charts, bisystems and chart-bisystem cells form a pseudo double category with the obvious composition. So do the lenses, bisystems and lens-bisystem cells. Finally, there is a notion of 3-cell given by a box whose sides are 2-cells of each kind, so that the resulting diagram in (with isomorphisms on two sides) commutes. The lens-bisystem double category is (recall that ), with the action given by the functor (restricted to isomorphisms). The chart-bisystem double category is the result of taking , a pseudocategory in double categories, and applying where takes the horizontal category of a (strict) double category. The 3-cells, of course, are the 2-cells of . Analogously to the above, for each class of 1-cells, there is a double category with those as the objects, the two types of cell as the two morphisms, and the 3-cells as the 2-cells. All of these six double categories admit a symmetric monoidal structure. We can recover the ordinary category of (Moore) systems as the slice over , in the following sense: Eigil Fjeldgren Rischel 2025 5 6 https://erischel.com/efr-2M0M/ efr-2M0M /efr-2M0M/ Proposition There is a pullback diagram in In a double category , there is a "horizontal slice" 1-category having objects the horizontal maps and morphisms given by 2-cells that are identity on the left boundary, composed vertically. Similarly there is a "vertical slice". This is given by a similar pullback in ---we simply do this one level up. Note that Myers' construction of the double fibration in fact uses the vertical slice in this sense. This somewhat trivial observation means that any composition in which produces 2-cell under in fact produces a morphism of systems in the ordinary sense. Replacing with another object, we may regard the slices as further-parametrized versions of . There is essentially no difficulty in applying this to the Markov case: Eigil Fjeldgren Rischel 2025 6 20 https://erischel.com/efr-W413/ efr-W413 /efr-W413/ Definition Let be a stochastic dynamical systems theory. Consider the double category whose vertical category is and horizontal category is , with commutative squares as the 2-cells (this is the transpose of ). This carries an obvious symmetric monoidal structure, and acts on the double category of stochastic arenas via the functor . Denote by the symmetric monoidal triple category induced as in Definition by this data. The information contained in is much as above, although the complications involved in the double category of stochastic arenas remain present. Our bisystems are reminiscent of the energy-driven systems of Capucci, Lynch, and Spivak (Reference ). Indeed, their is essentially the bisystems in the (ordinary) doctrine of smooth dynamical systems. As we mentioned in the introduction, Shapiro and Spivak (Reference ) have previous developed a structure which consists of the bisystems for the discrete dynamical systems doctrine (i.e ). has a tremendous amount of structure coming from the representation of lenses in this doctrine as the category of polynomial functors, which can't be replicated for a general systems theory (and certainly not for a general stochastic systems theory). Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-9J8G/ efr-9J8G /efr-9J8G/ A Stochastic Dynamical Systems Theory of Smooth Manifolds In this section, as the title suggests, we construct a stochastic dynamical systems theory of smooth manifolds, with the usual tangent bundle. The main point is to construct a Markov category containing the smooth manifolds which is pullback-positive. We do this by considering the larger category of diffeological spaces. In order to make the topology work, we need to complicate the notion of diffeological space a bit, but having done so, we obtain a representable Markov category which is pullback-positive, and contains as a full subcategory of the deterministic maps. A kernel is a Markov kernel valued in Radon measures which is weakly continuous---so induces a linear map taking to the function on the spaces of continuous functions---and which furthermore smooth in the sense that this operation preserves the smooth functions. This Markov category of "smooth stochastic maps" may be of some independent interest. Eigil Fjeldgren Rischel 2025 5 2 https://erischel.com/efr-CBBU/ efr-CBBU /efr-CBBU/ Definition Let denote the full subcategory of spanned by the objects for each . Note that has finite products, and is generated by the object under finite products. Eigil Fjeldgren Rischel 2025 5 2 https://erischel.com/efr-E1SZ/ efr-E1SZ /efr-E1SZ/ Diffeological Space Definition A smooth space is a sheaf on in the standard topology of open covers. A smooth space is a diffeological space if, for each the map is injective. A diffeological space with underlying set is called a diffeology on , and consists of specifying which maps are smooth. We call these maps smooth plots. Given a subset there is an obvious canonical diffeology on given by taking the plots to be those functions whose image in is smooth. We call this the subspace diffeology. A morphism of diffeological spaces is called a smooth map. It is equivalently a function which carries smooth plots to smooth plots. Eigil Fjeldgren Rischel 2025 5 2 https://erischel.com/efr-DX6S/ efr-DX6S /efr-DX6S/ Diffeo-Topological space Definition A diffeo-topological space is a diffeological space equipped with a topology so that all the smooth plots are continuous. A map of diffeo-topological spaces is a smooth map (for the diffeology) which is also continuous (for the topology). Given a subset there is an obvious canonical diffeo-topology on given by taking the plots to be those functions whose image in is smooth, and equipping with the subspace topology. Eigil Fjeldgren Rischel 2025 5 2 https://erischel.com/efr-42SC/ efr-42SC /efr-42SC/ Lemma The category of diffeo-topological spaces admits all limits, given by taking the limits in topological spaces and diffeological spaces (which have the same underlying set). Eigil Fjeldgren Rischel 2025 5 2 https://erischel.com/efr-7RS2/ efr-7RS2 /efr-7RS2/ Proposition Let be a diffeo-topological space whose underlying space is Tychonoff. Then the space of probability measures has a canonical diffeology given by those plots so that for each continuous function on the resulting map is continuous, and if is smooth, then this is smooth as well. With this diffeology, and the topology of weak convergence, is a diffeo-topological space. This defines a commutative affine monad on the category of such diffeo-topological spaces. Eigil Fjeldgren Rischel202552Proof The topology of weak convergence on is such that is continuous if and only if the expectation map carries continuous functions to continuous functions. But this is part of the requirement to be a smooth plot, so certainly this is a diffeo-topological space. Since the linear operator associated to under the unit is merely evaluation at the unit is clearly smooth. Consider . To test this is smooth, let be a plot. We must show is a plot. So let be a continuous (resp. smooth) function. We must show its expectation a is continuous (resp. smooth) function of . By construction is continuous (resp. smooth), and so since is a plot, the map is continuous (resp. smooth). But this is exactly what we wanted. The monad laws follow from their holding in . Since a commutative monad is equivalently a strong monad satisfying a certain equation (which holds for this monad in and therefore also here), it suffices to show that the strength is smooth. This follows by a completely analogous argument. Eigil Fjeldgren Rischel 2025 5 2 https://erischel.com/efr-S87B/ efr-S87B /efr-S87B/ Corollary The category of Tychonoff diffeological spaces and Kleisli maps for the monad described in Proposition is a pullback-positive Markov category. Its deterministic category is . There is a fully faithful functor which preserves transverse pullbacks. The maps between smooth manifolds are given by weakly continuous families of Radon probability measures, so that the expectation operator carries smooth maps to smooth maps. Eigil Fjeldgren Rischel 2025 5 2 https://erischel.com/efr-LPX2/ efr-LPX2 /efr-LPX2/ Remark Since we describe properties of kernels in terms of their corresponding linear operator on function spaces, it would seem natural to take the function spaces as the basic object. Hence we might consider the category -algebras with some relaxed notion of maps between them. The tricky part there is to find some reasonable class of maps so that the tensor product (coproduct) of -algebras extends to these. (Since it is not the same as the tensor product of -algebras, linear maps do not automatically extend to the tensor product). In particular, when considering kernels where is a "fat point of order 1"---that is, functions on have a value at the point and a derivative---it is not clear what sort of continuity condition the derivative operation on should satisfy, nor how to define this in a general way for all -algebras. It would certainly be of interest to synthetic computational geometry to have such a Markov category, but we leave this for future work. We will now give an example of how to represent the training dynamics of a machine learning system using the tools developed so far. As discussed previously, given a parameterized function , its reverse derivative naturally becomes a parameterized lens, and the composition of these describe how gradient vectors are passed around to compute an update during training. It is natural to want to compose this lens with the data-generating distribution , (along with some more context describing the loss function, etc) to obtain the training dynamics of such a model. This requires a category of parameterized lenses which allows stochastic maps in the base. The goal of combining this feature with non-trivial tangent bundles was one of the original motivations for developing a theory of stochastic lenses. Eigil Fjeldgren Rischel 2025 5 3 https://erischel.com/efr-IMZ3/ efr-IMZ3 /efr-IMZ3/ Example Consider the Markov prefibration . Equip this with the section - this described discrete-time systems (whose update is required to be smooth in the input and present state). This is clearly a symmetric monoidal functor and thus defines a systems theory. Note that this is completely different from the ordinary tangent bundle, despite the coincidence of notation. Let and be two bisystems in this theory. As in § , we may define a parameterized lens , denoting by the coproduct in lenses, and by the Markov structure defined in Corollary . Observe that is simply the indexed set . There is an obvious indexed map from this to , given by and . In words, we receive an update either to the -state or to the -state. We apply this update to the relevant state and leave the other alone. This defines a lens , which we may compose with the above to obtain a bisystem . Let us denote this by . This has very much the same flavor as the external choice for open games, although we will not develop the theory of this operation in detail here. Now, given a smooth (deterministic) map , where are smooth manifolds (not merely diffeo-toplogical spaces), we obtain a lens , where denotes the cotangent bundle. (This does not, prima facie, make sense for a general diffeo-topological space). Let us take as given some family of lenses for various . Such an operation amounts to choosing a way of updating given a cotangent vector---hence we can see it as an optimization algorithm. One example of such would be gradient descent, which given a choice of Riemann structure on , takes a step of a given length in the direction which most quickly decreases the given covector. (It should be noted that there are more complicated optimization strategies which don't fit this particular pattern - for example, momentum algorithms have to maintain some extra internal state other than . But let's stick with this pattern for this example). Note also that we're not assuming the optimizers are a natural transformation or anything like that. Now we are ready to build the neural network architecture known as a generative adversarial network, or GAN (Reference ). Let us first describe the idea. Our goal is to generate additional samples from some distribution, given a set of existing samples---for example, our goal may be to generate more pictures in the same style as an existing corpus. Suppose our data is of type , and let be the data distribution. We fix some latent distribution , where is any space of our choice---usually, is and is a Gaussian. Finally we choose two neural networks, the generator , and the discriminator . The goal of the discriminator is to discriminate real samples from the data from generated samples, by providing a low value on the generated samples and a high value on the true samples. The training process now goes as follows: for each step of training, we either sample from the latent distribution, and have the generator use this to generate a sample, or draw a sample from the data distribution (choosing between these with some probability ). Then in either case, we have the discriminator score the generated sample. If the sample was generated by the generator, the discriminator's loss is equal to its output, otherwise it is equal to times its output. We update the discriminator according to the gradient of this loss (minimizing it), and update the generator (if we are in the branch where it was run) according to the negative of the gradient of this loss with respect to the generator parameters---this amounts to doing a gradient descent update on the generator for the negative of the discriminator loss. We can represent this schematically using the following tape diagram (see Reference ): ] \tikzstyle {Tape boundary}=[-, draw={rgb,255: red,191; green,0; blue,64}] \resizebox {\textwidth }{!}{ \begin {tikzpicture}[scale=0.5] \begin {pgfonlayer}{nodelayer} \node [style=State] (0) at (-5.5, -2.5) {Data}; \node [style=State] (2) at (-10, 5.25) {Latent}; \node [style=System] (4) at (-4.5, 5.25) {Generator}; \node [style=State] (6) at (-4.5, 3) {+1}; \node [style=State] (7) at (-5.5, -5.25) {-1}; \node [style=System] (8) at (4.75, 0.25) {}; \node [style=System] (9) at (4.75, 0.25) {Discriminator}; \node [style=System] (10) at (9.25, 0.25) {Loss}; \node [style=none] (11) at (-3.75, 5) {}; \node [style=none] (12) at (-1.75, 4.75) {}; \node [style=none] (13) at (1.5, 0.25) {}; \node [style=none] (14) at (3.25, 0.5) {}; \node [style=none] (15) at (5, 0) {}; \node [style=none] (16) at (5, 0.5) {}; \node [style=none] (17) at (9, 0) {}; \node [style=none] (18) at (9, 0.5) {}; \node [style=none] (19) at (6.5, -1) {}; 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\draw (32.center) to (78); \end {pgfonlayer} \end {tikzpicture}} ]]> The two "tapes" branching off at the start represent two maps (bisystems) composed by , while the backwards wires indicate the flow of the gradients. The ground symbol indicates a value being discarded. Note that the theory of tape diagrams has only been developed formally for distributive categories, and for essentilly the same reason as in § , we do not have distributivity in this case. However, the interpretation of this diagram is still unambiguous---distributivity is required to make tape diagrams complete, not to make them sound. Concretely, if we tried to represent the tensoring of this system with another system, we would have no way of doing it, but if the category was distributive we could do so by adding this additional system to each of the branches. Still, the figure is best viewed as a visual aid rather than a formal representation. Eigil Fjeldgren Rischel 2025 3 26 https://erischel.com/efr-O088/ efr-O088 /efr-O088/ Markov Fibrations and Stochastic Lenses Eigil Fjeldgren Rischel 2025 4 7 https://erischel.com/efr-X0ZV/ efr-X0ZV /efr-X0ZV/ Introduction Let denote the Kleisli category of the discrete (countable) distribution monad on . This is a simple setting for working with probability theory---sufficient for many applications. In order to study compositional Bayesian game theory (Reference , Reference ) one studies the category of optics in . These have the right level of expressivity to talk about players taking random actions, and where payoff depends stochastically on players' decisions. In . Games with this codomain naturally describe the situation of a player who has a binary choice between or . We call coproducts of this form the "good" coproducts---note that also , but this is considered somewhat pathological, since it relies on the nonexistence of any morphism when and are nonempty. In order to describe this structure on it would be useful if it had all coproducts. Unfortunately this is not the case. has a well-known extension with all coproducts, given by the fiberwise opposite of the fibration (this is Proposition in the case ). Extending this to stochastic maps would be the obvious way of constructing such a category of "dependent optics". Consider a category defined as follows. Its objects are the objects of ---that is, they are indexed families of sets. A map in is a stochastic map in the base , and a stochastic map on the total spaces which is compatible with it. Note that if is surjective, the map on the base is fully determined by the map on the fibers, which must merely satisfy the condition that the distribution of the indexing point in depends only on the indexing point in , not the specific point in the fiber . We claim is a reasonable notion of "stochastic charts". Recall that by "chart" we mean something like "lenses where both maps go forward". If stochastic lenses are supposed to include optics as the full subcategory spanned by the "non-dependent" objects, then the charts should include "co-optics"---that is, maps between and should be given by the coend And in fact this is the case: Clearly there is a map from this coend to maps in . By taking and conditioning on , we see this is surjective. Finally, by restricting to the support of the forwards part inside , we obtain a representative for each element of the coend which is uniquely determined by and (since the conditional is well-defined on the support). Note: This relies both on the fact that has conditionals, and on the existence of supports. We've previously seen these defined in abstract Markov categories (Definition )--supports in of a morphism are simply given by those so that 0]]>. Note that the existence of both conditionals and supports is a very strong assumption---the only categories we are aware of with both properties are those whose probability distributions have a discrete character, like and . Neither will be essential to the theory, but both will play a role in certain theorems---we will see more of this later. The goal of the theory of Markov fibrations is to give a notion of "fiberwise opposite" which can be applied to the codomain functor to give a reasonable notion of "stochastic lenses". In particular, we should recover the usual category of optics in the previous case. It is clear that the codomain functor is not a (Grothendieck) fibration, since this would require to have pullbacks, which can only hold for a Cartesian Markov category. However, we can do some things. Namely, given a Cartesian (pullback) square in and a map where the base map is deterministic, for each deterministic factorizing map there is a unique lift . In other words, the pullback over is a fibration---in fact, it is simply the family fibration . Furthermore, if is itself deterministic, there is such a unique lift even without assuming that the factorization is deterministic. Moreover, we can factor any map in as such an induced map followed by a map over a deterministic base, as follows: This gives us a hope that we can, in some way, control the category using the pullback over , which is a fibration, and some information somehow given by these extra maps. Note also that the diagram above is equivalent to giving: a span and a section , which all lives in the base, and a map in the fiber over . Thus it would seem to be very amenable to fiberwise dualization. Analogous to our argument above that "co-optics" are equivalent to maps in we can do the following: Suppose given two tuples as above. Suppose there exists a map over , so that . Then there is a canonical map over , because pullbacks commute. If the triangle moreover commutes, then these two triples represent the same map in These equivalence relations correspond to "sliding" for deterministic morphisms . Note that the condition here can be checked just on the fibration . To obtain the full set of sliding equations, we will need to use stochastic maps , and thus leave that fibration behind. However, we are tantalizingly close to realizing as being presented by some sort of additional structure on the fibration . (For a general monad acting on , the category , of effectful optics up to sliding of pure morphisms, was studied by Riley in Reference , section 4.9, and by Hedges in Reference ) In this chapter we will indeed provide such a structure, and analyze it. In § , we'll axiomatise the lifting property of discussed above into a property we call a Markov prefibration (Definition ). In § , we exhibit a free Markov prefibration associated to a fibration---its morphisms are precisely spans of the form seen above. Naturally, given a prefibration , its underlying fibration on becomes an algebra for the monad of this adjunction. In § , we characterize the class of prefibrations which are presented by their underlying algebra in this way---these are the Markov fibrations (Definition ). Since the monad commutes with fiberwise opposites, this yields a notion of fiberwise opposite for Markov fibrations. Following this, we review a few properties of the theory of Markov fibrations, including the existence of coproducts in the fibration (Proposition ), the stability of Markov fibrations under limits (§ ), and induced monoidal structures (§ ). Combining these, we can prove: Eigil Fjeldgren Rischel 2025 4 27 https://erischel.com/efr-OSH4/ efr-OSH4 /efr-OSH4/ Theorem There exists a (strong) symmetric monoidal functor which Is fully faithful. Preserves the good coproducts. So that the image has all coproducts, and every object of the image is a coproduct of objects in And so that moreover the monoidal structure on the image is distributive Eigil Fjeldgren Rischel 2025 3 26 https://erischel.com/efr-2IMZ/ efr-2IMZ /efr-2IMZ/ Markov Prefibrations Eigil Fjeldgren Rischel 2024 11 21 https://erischel.com/efr-0045/ efr-0045 /efr-0045/ Remark Markov categories generally do not have pullbacks, for the same reason that they usually don't have products. This issue generally hinders the construction of fibrations, in the ordinary sense, of Markov categories. However, we can go part of the way. The idea of the following definitions is that given a pullback in the deterministic category, say , a map where the -coordinate is deterministic should be uniquely determined by a choice of (deterministic) map and (stochastic) such that the square commutes---as we claimed above (and will see below), this holds for the Markov category of discrete probability . The analogous statement for products---that a map with deterministic -component is uniquely determined by the projections (or marginals) ---is a consequence of positivity (see Proposition ), and hence holds in most Markov categories of interest. Eigil Fjeldgren Rischel 2024 6 24 https://erischel.com/efr-0019/ efr-0019 /efr-0019/ Markov prefibration Definition Let be a Markov category, and let be a functor into it. Then we call a Markov prefibration if the following two conditions hold: The pullback is a (Grothendieck) fibration Given maps , in , such that are deterministic and are Cartesian for the above fibration, induces a bijection between maps such that , and maps so that . Note that when restricted to those maps where is deterministic, this being a bijection is the defining property of being Cartesian (for any , not necessarily a Cartesian one). Given a Markov prefibration we write for . We will refer to this as the deterministic part of ---note that this does have the potential for confusion, as when is itself a Markov category, this is not necessarily the same as the deterministic subcategory of . When lies inside , and is Cartesian for that fibration, we will simply refer to it as a Cartesian map in (there are no other types of Cartesian maps, so this should not lead to confusion). A morphism of Markov prefibrations is a functor over which preserves Cartesian maps. The category of Markov prefibrations over thus defined is denoted . Taking the deterministic part defines a functor . Eigil Fjeldgren Rischel 2025 4 6 https://erischel.com/efr-EQ41/ efr-EQ41 /efr-EQ41/ On 2-category theory Remark Since we will shortly be working with a number of functors between categories whose objects are themselves categories with some structure, it may be thought that we should give some consideration to the strictness of our constructions---for example, we will shortly construct a left adjoint to and it may well be asked how strict this adjoint is, whether we need to consider the definition of pseudomonad when we get so far, et cetera. However, we can largely avoid this issue. The key observation is that none of our functors will alter the objects of the underlying category (since is identity on objects,). Hence, all the natural transformations that we would ordinarily ask to be equivalences of categories will instead be isomorphisms, and we can largely ignore considerations of higher category theory---similarly, all our functors will be strictly functorial. As a simple example of this, observe that the pullback functor is automatically strict---it simply consists in restriction to a subset of the morphisms in (which is automatically closed under composition), and thus clearly preserves composition strictly. Eigil Fjeldgren Rischel 2025 4 29 https://erischel.com/efr-DINO/ efr-DINO /efr-DINO/ Definition In a Markov prefibration (as previously noted), a morphism in is called Cartesian if is deterministic and is Cartesian in the fibration . It is called vertical if is an identity. It is called a stochastic-Cartesian if there exists a Cartesian map so that (recall that in this case is uniquely determined by and ). Note that if is stochastic-Cartesian and is deterministic, then is Cartesian. The introduction to this chapter contains the argument that is a Markov prefibration. This is a key motivating example. Eigil Fjeldgren Rischel 2025 4 8 https://erischel.com/efr-CJTH/ efr-CJTH /efr-CJTH/ Markov prefibrations over Cartesian base Example Let be a Markov category which is Cartesian (that is, one where all morphisms are deterministic). Then a Markov prefibration over is simply a Grothendieck fibration. Eigil Fjeldgren Rischel 2024 11 19 https://erischel.com/efr-0044/ efr-0044 /efr-0044/ Proposition Let be a Markov category. Then is a Markov prefibration with Cartesian maps given by pullback squares, if and only if is pullback-positive. Eigil Fjeldgren Rischel20241119Proof Assume first is pullback-positive. It is clear that computing pullbacks in gives the required Cartesian lifts---pullback positivity is precisely the claim that lifts exist uniquely in the definition of a Cartesian morphism (since the map to the base leg of the pullback is always deterministic in this case). Since Cartesian lifts can be taken to be deterministic (we have just constructed deterministic Cartesian lifts, and such lifts are unique up to unique isomorphism), the second condition also follows from this assumption, simply taking the other leg to be deterministic. Conversely, suppose is a Markov prefibration and suppose the Cartesian maps are given by the deterministic pullback squares. Let be an object of and let be a deterministic map, and form the pullback , which is Cartesian. Let be deterministic and let be any map. Expanding the latter into a map from the object to the Cartesian property of the square implies there is a unique pairing Eigil Fjeldgren Rischel 2024 11 19 https://erischel.com/efr-0040/ efr-0040 /efr-0040/ Remark In a general Markov category, not every isomorphism is necessarily deterministic. This means that, in general, fibres over isomorphic objects in a Markov prefibration are not necessarily isomorphic or even equivalent as categories. This seemingly immoral situation is, in fact, in accordance with other results indicating that deterministic isomorphism is really the proper notion of identification in a Markov category. See eg Reference , Section 4, for further discussion of this point. (Since the basic idea of a Markov category involves objects equipped with some structure which is not preserved by all the morphisms, it is not so paradoxical that an isomorphism in this situation should be insufficient to render two objects identical). Under very weak assumptions on the Markov category , such as positivity, all isomorphisms are deterministic. This implies that all Markov prefibrations over are isofibrations, and thus rules out any sort of behavior like the above. As noted, we are only concerned with positive Markov categories. Relatedly, in the proof of Proposition , a careless prover may have erroneously concluded after the first step that all Cartesian lifts are deterministic---but since Cartesian lifts are characterized only up to isomorphism, not necessarily deterministic isomorphism, this does not automatically follow. (But of course replacing a nondeterministic lift with an isomorphic deterministic one in this situation cannot alter the unique existence of the factorization, so it does not matter). Eigil Fjeldgren Rischel 2024 11 19 https://erischel.com/efr-0042/ efr-0042 /efr-0042/ Proposition Let be a Markov prefibration, let be full subcategories so that is contained in , and suppose is a monoidal subcategory (which is then automatically a sub-Markov category). Suppose finally is stable under pullback along deterministic morphisms in . Then is again a Markov prefibration. Eigil Fjeldgren Rischel20241119Proof By assumption, given and , the Cartesian lift is again in . The fullness of the subcategory inclusions suffices to prove the existence and uniqueness of the required lifts so that this is still Cartesian after restricting. For the same reason, since we have just observed that the Cartesian lifts are the same as in , the second part of the definition also holds. Eigil Fjeldgren Rischel 2025 3 27 https://erischel.com/efr-NC7D/ efr-NC7D /efr-NC7D/ as Markov prefibration Example Recall that the deterministic maps in are those kernels valued in -valued measures. (And that these are not the same thing as the measurable maps). Given a pair of such maps, we claim that the subset of given by those points where as measures on , equipped with the subset -algebra and its inclusions into and , is a pullback in . To see this, observe that is the Kleisli category for the submonad of the Giry monad which takes a space to the subspace of -valued probability measures on it. Note that by the general theory of Markov categories this has products given by the products in . It hence suffices to prove that the pullback of along the diagonal exists---if it does, it has the universal property of the product . To see that it does (and that it's given by the space described above), note that the diagonal is a split monomorphism, so it suffices to show that factors over if and only if the composite to lifts over the diagonal. Now this lift over exists if and only if the probability of the diagonal under the composite is . By definition, it is Since the function being integrated is an indicator, this is simply the measure of the set If the probability of the diagonal is one, clearly the two marginals agree. Conversely, since is Cartesian, it must be the case that if the marginals agree the probability of the diagonal is 1. Therefore this is equal to the subset described at the start. But for this to have measure for all is equivalent to the desired lifting property. This finishes the argument. Moreover, this argument relied only on the -valuedness of the maps , not or its marginals. Hence this also proves that the pullback along the diagonal is preserved by the inclusion . Since is known to be positive, this implies is a Markov prefibration. Eigil Fjeldgren Rischel 2024 11 19 https://erischel.com/efr-0041/ efr-0041 /efr-0041/ Proposition The codomain functor is a Markov prefibration. Eigil Fjeldgren Rischel20241119Proof We wish to apply Proposition . The only thing to check is that standard Borel spaces are stable under pullbacks in . But standard Borel spaces are known to be stable under products and measurable subsets, and this is enough (see eg. Reference propositions 3.1.23 and 3.3.15) Eigil Fjeldgren Rischel 2024 11 19 https://erischel.com/efr-0043/ efr-0043 /efr-0043/ Corollary is a Markov prefibration. Eigil Fjeldgren Rischel 2025 3 29 https://erischel.com/efr-8B5X/ efr-8B5X /efr-8B5X/ Example The functor is not a Markov prefibration, although its pullback over is a Grothendieck fibration. To see this, first consider the deterministic pullback. An optic with deterministic base can be identified with a map (and the base deterministic map ). To see this, first observe that the subset of with the marginal deterministic is in bijection with , since is positive. Hence we can calculate using the ninja yoneda lemma as in Proposition . Hence this part is a fibration with the fiber over being the coKleisli category of the monad, and the pullback functors given by reindexing these parametrized maps. The Cartesian lift of a map at is given by the optic with unit residual and identity backwards component. Now, let denote the standard Gaussian distribution, let be the function given by if and otherwise, and consider the two optics given by (where we recall that the first argument is the residual). They cannot be equal, as postcomposition with the optic given by the identity yields, for the former, the standard Gaussian , and for the latter, the constant zero map. But postcomposition with the projection does give the same optic (the identity), because, for every fixed , when is normally distributed, with probability one. Hence the unique lifting of Cartesian maps over Cartesian maps cannot hold. This counterexample indicates that, although is a Markov prefibration, we can not expect a dual version of this prefibration---in fact, since over deterministic maps is the fiberwise dual of (the restriction to trivially-indexed objects of) this example shows that there is no Markov prefibration whose deterministic part is the fiberwise dual of . Moreover, as the example indicates, this is not a mere technical issue, but an unavoidable fact about optics in general measurable spaces---even up to behavioral equivalence, they simply don't satisfy the conditions of being a Markov prefibration. (But see Theorem ) On the other hand, we have: Eigil Fjeldgren Rischel 2025 4 3 https://erischel.com/efr-UDS9/ efr-UDS9 /efr-UDS9/ Proposition Let be a positive Markov category with supports. Then is a Markov prefibration. Eigil Fjeldgren Rischel202543Proof As above, we see that the deterministic part is a fibration, so take deterministic maps, and let be an optic so that the induced triangle with the two Cartesian lifts to commutes. Let the two parts be . The implication is that -almost surely, is equal to the projection to (and renders the triangle in commutative). Note that factors over the support of this map, hence we can assume the marginal has full support. Hence up to sliding equivalence, is strictly equal to the projection. This implies the lift is uniquely determined as desired. The existence of supports rules out the pathological behaviour. Essentially, in the presence of supports, we can sensibly reason about "the points of measure zero" and exclude them from consideration---and Proposition implies that the independent pairing of two measures always have the least "points of measure zero", and so that what can be proven equivalent under the assumption of independence will always be equivalent. By contrast, the map from Example satisfies for almost all when is normally distributed, for all , but this does not imply that for all measures on with this marginal, is distributed as the marginal of . One point of view is that the map is simply pathological, and we should restrict our attention to maps that are continuous in some sense (from the point of view of computer science, one argument for this is that computable maps are necessarily continuous). The category of Tychonoff spaces and weakly continuous kernels does indeed have supports. However, since it lacks conditionals, it is still not ideal from our point of view. Eigil Fjeldgren Rischel 2025 3 26 https://erischel.com/efr-GO6R/ efr-GO6R /efr-GO6R/ Free Markov Prefibrations Because every map in factors into arrows which are "induced" from arrows in and the Markov prefibration property, it may initially be hoped that is in some sense "free" on the data of the fibration and the inclusion . If that was true, we may further hope that taking the fiberwise opposite of the fibration and applying the same free generation principle would generate a good notion of stochastic lens. Unfortunately, this is not the case. We will see that the free prefibration is given by gadgets which look a bit like an indexed version of optics, up to a sliding equivalence for deterministic maps on the residual. This prompts us to look for some extra structure on the fibration which describes sliding equivalences for stochastic maps on the residual. In the next section, we will see that this is exactly the structure of an Eilenberg-Moore algebra for the free prefibration monad on . In this section, we will give a description of the free Markov prefibration on a fibration (assuming is pullback positive). There is a fairly simple description of the hom-sets, but their composition is a bit tricky, and verifying associativity even more so. Hence we will employ a technical trick: by characterizing the hom-sets as "freely generated" in a certain sense from the hom-sets in , we can identify them with sets of natural transformations using a Yoneda-type argument, and infer composition and associativity from there. Eigil Fjeldgren Rischel 2024 6 25 https://erischel.com/efr-001I/ efr-001I /efr-001I/ Indexed copresheaf Definition Let be a functor. An indexed copresheaf on is a tuple consisting of a copresheaf, an object of , and a natural transformation as indicated. We say the indexed copresheaf is over , and we will abuse the terminology by referring to itself as an indexed copresheaf, leaving the transformation implicit (for example, "let be an indexed copresheaf over "). We denote the subset for by . A map of indexed copresheaves is a natural transformation and a map so that the obvious square of natural transformations commutes. We denote the category of indexed copresheaves by . Note that there is an obvious forgetful functor Observe that for each object , there is a corepresentable copresheaf . Maps between these obey the Yoneda lemma, in the sense that they are in bijection with maps between the underlying objects in . This defines a fully faithful functor over . Of course, there is a dual notion of indexed presheaf, but this will not interest us. Eigil Fjeldgren Rischel 2025 3 27 https://erischel.com/efr-FOJT/ efr-FOJT /efr-FOJT/ Proposition Let be any functor and let be an identity-on-objects functor. Write for the pullback. If is a copresheaf indexed over , the pullback is a copresheaf on indexed over again in a unique way. This defines a functor . Moreover, this functor preserves the corepresentable copresheaves (since is identity on objects, so is , so this statement makes sense). Eigil Fjeldgren Rischel 2025 3 27 https://erischel.com/efr-FRQG/ efr-FRQG /efr-FRQG/ Corollary Let be a functor and let be identity-on-objects and faithful. Then the pullback of the composite and the inclusion is identical to . Therefore, Proposition gives a functor . The idea of our construction of the free Markov prefibration is to give a certain monad on and consider the Kleisli maps between the representable copresheaves. At this point, the notion of a deterministic map equipped with a stochastic (ie not necessarily deterministic) section begins playing a key role. The phrase "stochastic section" will always carry an implicit "of a deterministic map". In most cases the map that the section is a section of will be clear from the context. Note that, given a stochastic section and a deterministic map , if is pullback-positive, there is a unique lifting of this to a section of the projection . We will use this fact several times. Eigil Fjeldgren Rischel 2025 3 27 https://erischel.com/efr-Y926/ efr-Y926 /efr-Y926/ Stochastic Module Definition Let be a Markov category, and let be a fibration. Then a stochastic module consists of An indexed copresheaf For each , Cartesian morphism lying over , and stochastic section , a function , which acts on the underlying morphisms in as composition with Satisfying, whenever given a commutative square of Cartesian morphisms: and stochastic sections lying over so that we have a digram in : Where the maps except are deterministic, and are sections of and , and both the square of deterministic maps and the square involving commute, the condition that the square commutes. And satisfying furthermore the equation, for every two stochastic sections the equation (note that this makes sense because pullbacks compose). A morphism of stochastic modules is an indexed natural transformation which preserves the operations . The category of stochastic modules is denoted . There is an apparent forgetful functor . Eigil Fjeldgren Rischel 2025 4 27 https://erischel.com/efr-GL6R/ efr-GL6R /efr-GL6R/ Remark Note that of course depends on , not just . If is a deterministic map, is a stochastic section, and are two Cartesian lifts of to , then applying the commutativity axiom for stochastic modules implies that the triangle commutes. In what follows, we will simply write for some choice of cartesian lift, and speak of . The above shows that this is a harmless abuse---the actions are preserved by the identification of different Cartesian lifts. In particular, we will often make arguments as if pullbacks compose strictly, although in general they only compose up to isomorphism. The above triangle means this is harmless. The term "stochastic module" is not very good, but this is mostly a nonce definition in any case, so we won't worry too much about it. Stochastic modules over a given can be seen to be monadic over the category of indexed copresheaves over that . However, the compatibility of these local left adjoints with the structure of the rest of the category is somewhat subtle. However, we do have free stochastic modules on representable indexed copresheaves, as we will soon see. Eigil Fjeldgren Rischel 2025 3 27 https://erischel.com/efr-2XPE/ efr-2XPE /efr-2XPE/ Proposition Let be a Markov prefibration, and let . Then the corepresentable copresheaf , restricted to , (but not pulled back---that is, we remember the whole set , even the part over stochastic , but only the composition with maps in ) is a stochastic module in a canonical way, with given by composition with the unique induced lift of . Moreover, any morphism of Markov prefibrations induces a homomorphism of stochastic modules Eigil Fjeldgren Rischel2025327Proof Given and a stochastic section , it's clear that composition with the unique lift is a map of the right type, so we just have to verify the equations. For the first equation (item 3 in the definition of stochastic module), we are comparing two maps . These are given by composition with two maps, let's call them . These two maps are lifts of and , but by assumption these two are equal. Hence by the uniqueness property of Markov prefibrations, , and we have our equation. The other equation follows in a completely analogous way. To prove the homomorphism property, note that a morphism of prefibrations preserves Cartesian morphisms, and hence (by uniqueness) must preserve the unique lifts of stochastic sections. Then by functoriality it must preserve composition with these, which finishes the proof. Eigil Fjeldgren Rischel 2025 3 27 https://erischel.com/efr-OH7U/ efr-OH7U /efr-OH7U/ Lemma Let be an object. Then there is a free stochastic module on its representable copresheaf, in the sense that if is another stochastic module, homomorphisms are in bijection with indexed natural transformations The free stochastic module on a corepresentable presheaf is given as follows: an element of consists of a diagram in of the form (where ,) where and are deterministic, plus a map lying over . This is up to the equivalence relation which, given some other such tuple, identifies them whenever there exists deterministic as in this diagram: so that the two deterministic triangles commute, and so that the unique Cartesian map over forms a commutative triangle with the two maps to . (Note that we do not claim the relation just described is inherently an equivalence relation, rather we form the equivalence relation generated by this). It is clear how a map acts on this to make it a copresheaf. It is indexed by taking a tuple as above to the composite . Given with a stochastic section , and an element of the induced element in is given by forming the pullback taking the pullback of the map over to one lying over the projection and composing the section with the induced lift The underlying copresheaf of this respects Cartesian maps in , in the sense that given a Cartesian map and an element lying over a deterministic map the natural map from lifts to lifts is a bijection. Eigil Fjeldgren Rischel2025327Proof First, we have to verify the equations of a stochastic module for . For the first case, suppose we are given a square with all but the upwards maps stochastic. Now let be some object over and suppose we are given an element of , represented by a span , a section and a map over . Consider then the below diagram: By definition, the first of the two possible elements of are given by either forming the pullback , taking the lift of to and composing to get a section , and pulling back along the projection to get a map , then taking the span . The second is given by first composing with the map , then applying the above procedure with the pullback . The induced map exhibits the equality of these two under the equivalence relation defining (commutativity of the bottom-right square implies that triangle of stochastic sections commutes.) Given some other stochastic module over with a map of indexed copresheaves over , there is at most one extension to a map of stochastic presheaves over ---given an element with representative it must go to the identity element of acted on by the stochastic section to produce an element of followed by applied to the map over . But it is not hard to see that the equivalence relation imposed by is implied by the equations of a stochastic module, and so this map is well-defined, establishing the property. Secondly, let be a Cartesian map over and take a commutative triangle of deterministic maps. Finally take an element of over the given map . We must show it has a unique lift to over the given map . Let us take a representative given by a diagram: First, note that the two maps and do not necessarily agree. However, we can remedy this by replacing by their equalizer---note that as we argued above, this gives an equivalent element of the stochastic module. (By writing their equalizer in as the split pullback we can see that the section factors over this, even if does not have all equalizers in general). Hence we can assume the triangle formed by adding the dashed arrow commutes. Since now factors over , the lift gives a lift . Now by the Cartesian property of , there is a unique lift of to this map (here we just use the fact that is a fibration). This gives the desired lift. Finally, given two distinct lifts (again, we can assume their maps factor over ,) clearly any map witnessing an identity between their composites would likewise exhibit an identity between their lifts (since pullbacks compose). This proves uniqueness, and finishes the proof. We are now ready to prove the main proposition of this section: Eigil Fjeldgren Rischel 2025 3 27 https://erischel.com/efr-LGOA/ efr-LGOA /efr-LGOA/ Proposition Let be a pullback-positive Markov category and let be a fibration. Consider the full subcategory of stochastic modules spanned by the free modules on the corepresentables. Denote the opposite of this category . Clearly there is a commutative diagram We claim: is a Markov prefibration. There is a bijection . When the left-hand side is equipped with the canonical stochastic module structure, and the right is equipped with the free one, this is moreover a homomorphism (hence isomorphism) of stochastic modules. is initial among Markov prefibrations receiving a map from . In other words, this construction gives a left adjoint to the pullback functor Eigil Fjeldgren Rischel 2025 3 27 https://erischel.com/efr-0BES/ efr-0BES /efr-0BES/ Proof First observe that, by Lemma , the pullback is indeed a fibration, with the image of the Cartesian lifts under the functor being Cartesian again. (This also establishes that really does receive a map of fibrations from ) Given Cartesian , and a stochastic lift consider the pullback , and the pullback of to it. There is a unique lift of to a section , and this induces a unique lift using the stochastic module structure. The composite of this with the projection to is a lift of over , as required by a Markov prefibration. Analogously to the proof of Lemma , given some other lift , the fact this is a factorization implies the existence of some with maps and a lift of the section , so that the induced map between the pullbacks over and of makes the triangle into commute. But then since this is a triangle over deterministic bases, this implies the lifted triangle to also commutes, hence this lifts to another representative of the lift we started with. But then it's not hard to see that this maps to and exhibits an equation with the previously constructed "canonical" lift. Hence is a Markov prefibration, and by the above, the induced stochastic module structure on the corepresentable presheaves is exactly the one given by (in other words this equation is not merely a bijection of sets, but an isomorphism of stochastic modules). Let be a functor over into some other Markov prefibration which preserves Cartesian maps. Using the algebra structure on we see there is a unique extension of to which respects the stochastic module structure. By chasing the diagram around it's easy to see that this is functorial, and hence gives a map of Markov prefibrations---conversely, any such map extending must be a stochastic module homomorphism. Thus there is a unique functor, proving initiality. This characterization of the left adjoint makes it fairly easy to understand the induced monad on . Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-ACIV/ efr-ACIV /efr-ACIV/ Remark We will sometimes refer to the morphisms of as precharts. Taking the fibration as an example, it is not too hard to see that the precharts between and are representatives of co-optics . (To see this, note that any prechart is equivalent to one where the apex of the span has the form and the right leg is the projection to . Then the rest of the data is a map and a map , since the -coordiante of the latter map is determined by the span). In fact their equivalence relation is given by sliding equivalence for deterministic maps (i.e morphisms in ). The precharts in will be called prelenses. We will speak of the tuple representing a prechart just as a "decorated span (representing ...)". When part of the structure is understood, or can just be left abstracted, we will denote such a decorated span simply by or even just . It will be clear from context which part of the structure is being specified. Eigil Fjeldgren Rischel 2025 4 7 https://erischel.com/efr-A08L/ efr-A08L /efr-A08L/ Proposition Let be a fibration. Then the underlying fibration of the free Markov prefibration, has fiber over given by Objects are simply objects of A morphism consists of a deterministic , a stochastic section and a map , up to the equivalence relation generated by, whenever is deterministic and is a factorization of , identifying with . Given two such morphisms , their composite is represented by equipped with the section formed as the composite of and the lift of to the pullback, and the composite Given deterministic , the pullback is given on such a map by taking the pullback the induced section, and the pullback of the map along the projection It is immediately obvious that we have: Eigil Fjeldgren Rischel 2025 4 7 https://erischel.com/efr-ILIQ/ efr-ILIQ /efr-ILIQ/ Corollary The free Markov prefibration monad commutes with fiberwise opposites. In particular, algebra structures on are in bijection with algebra structures on and fiberwise opposites lifts to an involution of . Eigil Fjeldgren Rischel 2025 4 25 https://erischel.com/efr-OEFJ/ efr-OEFJ /efr-OEFJ/ Stochastic Module Fibration Definition Let be a pullback-positive Markov category. The adjunction induces a monad on . A module for this monad is called a stochastic module over (or, to distinguish it from the copresheaves of Definition , a stochastic module fibration). The category of stochastic module fibrations is denoted Let us try to understand the structure of a stochastic module fibration. It is easiest to understand in the case of a projection map . Suppose we have two objects over . Then we think of a map over as a map , that is a map parameterized by (this is literally the case for a codomain fibration). Given a stochastic section of , which amounts to a stochastic map , the stochastic module stucture picks out a new map , which is given over each point by choosing the parameter according to , then applying . The definition of the composite in Proposition , in these terms, tells us that given maps , and maps , the composite of and is equal to the map obtained by forming the parameterized composite and applying the independent pairing . This is of course how composition is supposed to work in a Markov category. Eigil Fjeldgren Rischel 2025 4 25 https://erischel.com/efr-VF6V/ efr-VF6V /efr-VF6V/ Lemma Let be a stochastic module over , and let be given, so that every map except is deterministic. Suppose the deterministic part of the diagram commutes, , is the induced section, and is a section. Let be two objects over . Suppose given a map . Then In particular, this operation depends only on . Moreover, it is functorial, in the sense that given a diagram with the downwards maps deterministic, Eigil Fjeldgren Rischel 2025 4 25 https://erischel.com/efr-U436/ efr-U436 /efr-U436/ Remark The operations associated to stochastic lifts are "functorial" in the sense that . However they are not functorial in the sense that To make sense of this, consider a simple case of a map in . Given two objects over (in ), a map is equivalent to a parametrized map . The operation consists in sampling this parameter according to the distribution ---but since composition in the fiber over is defined by copying the parameter, but composition in the fiber over (i.e just ) is defined by composing the kernels under conditional independence, these only agree if the distribution is assumed to be deterministic. Note that if either or is pulled back from a map (i.e, if they do not depend on the parameter ), the composition is preserved. By construction, two morphisms in represented by spans with apex are identified if there exists a zig-zag of spans (decorated with sections from the domain and morphisms in the fiber, satisfying equations, etc). We will now prove a lemma that allows us to cut this down to a smaller set in many conditions. We will need the following hypothesis: Eigil Fjeldgren Rischel 2025 4 15 https://erischel.com/efr-2MH5/ efr-2MH5 /efr-2MH5/ Weak conditionals Definition Let be a Markov prefibration. We say (or, abusing notation, ) admits weak conditionals if, given a Cartesian map and any map the existence part of the Cartesian condition holds---that is, every factorization admits a lift, although not necessarily a unique one. We say a Markov category admits weak conditionals if its codomain functor is a prefibration which admits weak conditionals---this is equivalent to requiring that it is pullback-positive, and that all deterministic pullbacks are carried to weak pullbacks by the inclusion (in other words, that they satisfy the existence part of the universal property even for pairs of nondeterministic maps). Observe that, if admits conditionals, it certainly admits weak conditionals: given a pullback and maps form a Bayesian inverse of with respect to the given measure, and use that to build a lifting , which gives ---then a diagram chase verifies that this map has the desired properties. Eigil Fjeldgren Rischel 2025 4 11 https://erischel.com/efr-TZ3Y/ efr-TZ3Y /efr-TZ3Y/ Lemma Suppose admits weak conditionals, and let be a fibration. Then two morphisms in , represented by commutative diagrams as well as , for , are equal if and only if there exists a span over , with a stochastic section lifting both the sections to , so that the pullbacks of to agree. Eigil Fjeldgren Rischel2025411Proof The relation here described clearly implies identity, and contains all the generating identities, so it suffices to show it is an equivalence relation. Reflexivity and symmetry are clear, so transitivity is the only issue. It suffices to show that, given a span and maps so that the triangles commute, and so that the two induced sections agree, and a map which pulls back to , we can find as above. To do this, take . Clearly the maps to are over , and by the existence of weak conditionals there exists a common lift of the sections to . By functoriality of pullbacks, the pullbacks of to agree. Eigil Fjeldgren Rischel 2025 3 29 https://erischel.com/efr-U4RA/ efr-U4RA /efr-U4RA/ Markov Fibrations Recall that, given a functor with left adjoint , there is a "standard resolution" of any object , given by the "cofork" , where the two parallel maps are the two possible applications of the adjunction counit. The adjunction is monadic if and only if this is always a coequalizer, in which case the -algebra corresponding to is ---conversely, given an algebra , there are two parallel maps (given by and the counit,) and the object in corresponding to this algebra is given by this coequalizer. Eigil Fjeldgren Rischel 2025 3 29 Example Consider the adjunction between the category of monoids and the category of sets. Given a monoid , consists of lists of elements in , and consists of lists of such lists. The two maps consist in either concatenating the lists, or replacing each list with its product. Clearly these two maps are coequalized by the product map . Moreover it's clear that two lists have the same product if and only if they are identified in this coequalizer (simply consider a singleton list-of-lists, which identifies any given list with the singleton corresponding to its product). As a generalization of this, if this coequalizer exists for every algebra, they form a left adjoint to the canonical functor . Since we have seen that the monad of free Markov prefibrations commutes with taking fiberwise opposites, we may hope that such a left adjoint exists---a simple argument shows that, if it is, it is fully faithful, and we may say that those prefibrations in the image are the "fibrations" and define their fiberwise opposite as the fiberwise opposite applied to their underlying algebras. Although it turns out to not be quite so simple, we will take this idea as our starting point. Eigil Fjeldgren Rischel 2025 3 29 Example Let be the category of topological groups, and let forget both the group structure and the topology. Clearly this is right adjoint to the free group in the discrete topology, and the monad of this adjunction is the free group monad, which we write for now. The canonical comparison functor just forgets the topology. Given a (non-topological) group, described by a map , we can form the diagram of topological groups . Here is the free, discrete group on and is the free discrete group on the underlying set of . Their coequalizer is simply equipped with the discrete topology, which is indeed the left adjoint to In what follows, we will denote the monad simply by to avoid too many complicated nestings of overlines and parentheses. Eigil Fjeldgren Rischel 2025 3 27 https://erischel.com/efr-HVUT/ efr-HVUT /efr-HVUT/ Markov Fibration Definition If is a Markov prefibration, we call it a Markov fibration if the diagram is a coequalizer in . Given an algebra of the free Markov prefibration monad, let be as above. We say presents a Markov fibration if the coequalizer of these maps in is a Markov prefibration. The terminology "presents a Markov fibration" is justified by the following proposition. Eigil Fjeldgren Rischel 2025 3 27 https://erischel.com/efr-MAUM/ efr-MAUM /efr-MAUM/ Proposition Let be stochastic module which presents a Markov fibration. Then the underlying algebra of the Markov prefibration obtained as the coequalizer of is isomorphic to , and in particular this prefibration is a Markov fibration. This correspondence determines an equivalence of categories between the full subcategory of spanned by the Markov fibrations, and the full subcategory spanned by those algebras which present a Markov fibration. Eigil Fjeldgren Rischel2025327Proof Let be an algebra as assumed, and let denote the coequalizer given. By general nonsense there is an induced functor which is moreover an algebra homomorphism---the claim is that this is an isomorphism. By Lemma , pullback to the deterministic part preserves these coequalizers, so this amounts to the claim that the diagram is a coequalizer. But this is true for any algebra of any monad (in fact, the unit gives a splitting of this coequalizer). By general nonsense the fibration associated to an algebra which presents a fibration forms a partial left adjoint to . This left adjoint, by the above, has its image inside and hence there is an adjunction . The preceding furthermore proves that the unit of this adjunction is the identity, which implies that the left adjoint is fully faithful---but by definition it is essentially surjective, finishing the argument. Let us briefly summarize the relationship between Markov prefibrations, Markov fibrations, and stochastic module fibrations at this stage. A stochastic module fibration is a (Grothendieck) fibration over , equipped with some extra structure involving the whole category . Given a deterministic map two objects , and a map , we can think of this as a map parameterized by the fibers . Given a stochastic section , the stochastic module structure picks out a map corresponding to choosing this parameter randomly according to . A Markov prefibration is a category over with a particular unique lifting property. In the above situation, it gives a unique lift of , corresponding to choosing according to and leaving the -coordinate unchangd. By composing this lift with , then with the Cartesian , we get a stochastic module structure on the part of lying over deterministic maps (which is also a Grothendieck fibration). Given a stochastic module structure, there is a way of generating a category over , by freely adding the lifts corresponding to a Markov prefibration, then quotienting by the relations implied by the stochastic module structure. This does not necessarily yield a Markov prefibration. A Markov prefibration is called a Markov fibration if it is presented by its underlying stochastic module in the above sense. Eigil Fjeldgren Rischel 2025 4 8 https://erischel.com/efr-YFV2/ efr-YFV2 /efr-YFV2/ Weak supports Definition Let be a stochastic module fibration. Let be a stochastic section, let be an object, and let be an endomorphism of its pullback. Observe that if there exists so that and factors over , then . We say has weak supports if this implication is an equivalence We will make use of the following lemma: Eigil Fjeldgren Rischel 2025 4 5 https://erischel.com/efr-L7L2/ efr-L7L2 /efr-L7L2/ Lemma Let be a parallel pair in , and suppose both are identity on objects. Suppose moreover this is a reflexive pair, i.e there is some so that . Then the coequalizer in is again identity on objects, and is given on hom-sets simply by the coequalizer of the parallel pair Eigil Fjeldgren Rischel202545Proof The only nontrivial part is to verify that composition is well-defined on the equivalence classes in . It suffices to see that post- and precomposition with a fixed morphism both preserve this equivalence relation. Take some , and . We must show that . But simply write and we are done. Clearly the other side follows by duality, finishing the proof. Eigil Fjeldgren Rischel 2025 4 5 https://erischel.com/efr-TBZZ/ efr-TBZZ /efr-TBZZ/ Construction of Proposition Let be a fibration equipped with a stochastic module structure. Consider the equivalence relation on which identifies two precharts if there exists a map over and a stochastic section of which preserves the section from , so that (note that this makes sense because pullbacks compose). Then: This equivalence relation respects composition, and so defines a category which we denote In , is a coequalizer diagram. In particular, presents a Markov fibration if and only if is a Markov prefibration (in which case is the fibration it presents) If has weak supports, is a prefibration Eigil Fjeldgren Rischel202545Proof Recall that the fibration has fibers whose morphisms are given by tuples (up to a certain equivalence relation). Forming the free Markov prefibration on this fibration, we find that the morphisms are given by diagrams equipped with a map . The two maps carry such a thing to first, the map resulting from forgetting and just composing with to get a span (and composing the sections to get a new section), and secondly, the map with apex obtained by using the stochastic module structure to push down into a map over . It is clear that this is equivalently the equation described in the theorem. This establishes points 1. and 2., since by Lemma we can compute such coequalizers hom-set by hom-set. Now we wish to prove that is a Markov prefibration given weak supports. Since by the coequalizer presentation, its deterministic part is isomorphic to , the fibration property is automatic. It remains to verify that, given a triangle in with the vertical and horizontal maps deterministic, an object and Cartesian maps , there exists a unique lift in . Such a lift is given by a diagram equipped with . By taking the equalizer of the two maps (the section factors over this), we may assume these two are equal, which implies that the pulled-back objects are equal---denote this object . Now the hypothesis is that after postcomposing with the Cartesian map this gives the map . This postcomposition is given simply by postcomposing the leg with the map (and observing that, by functoriality of pullbacks, this does not alter the pulled-back object). Then the claim is that integrating this map down into a map it gives the identity. But by assumption this means we can pull back to some object (lifting the section from ) where the two maps are already equal to the identity. But this pull-back can be applied to the original map as well. But this implies every such map is equal to the one represented by the diagram and the identity on with the map giving the witness, since identities pull back. This map only depends on the underlying hence is indeed a prefibration. It is not apparent whether weak supports are necessary for to be a prefibration. We have not found any counterexample, but in general the equivalence relation on charts is fairly complicated, so it is not apparent how to prove the necessity. We will generally not be too bothered about assuming weak supports instead of the more nebulous assumption that presents a Markov fibration. Eigil Fjeldgren Rischel 2025 9 13 https://erischel.com/efr-26SF/ efr-26SF /efr-26SF/ Lemma Let and be two representatives of charts. Given some possibly stochastic map over and , recall (Lemma ) that we can define , regardless of whether is deterministic or the section of a deterministic map. If there exists such a map , we can always factor it over the pullback as a section followed by a deterministic map. Hence the equivalence relation defining is equivalent to the relation identifying two representatives whenever there exists such an with Eigil Fjeldgren Rischel 2025 4 29 https://erischel.com/efr-XZRN/ efr-XZRN /efr-XZRN/ Remark By construction, for each Markov prefibration there is a canonical functor over , which restricts to an isomorphism on the deterministic part (and in particular preserves Cartesian morphisms). is a Markov fibration if and only if this is an isomorphism. Also by construction, given a morphism of stochastic modules there is an induced functor over . This restricts to on the deterministic part and in particular preserves Cartesian morphisms. Eigil Fjeldgren Rischel 2025 4 27 https://erischel.com/efr-1YS9/ efr-1YS9 /efr-1YS9/ Proposition Let be a stochastic module fibration. Then presents a Markov fibration if and only if does it. Eigil Fjeldgren Rischel2025427Proof Consider a triangle of this for in : where the maps to are deterministic. Suppose given Cartesian lifts of the cospan. These are the same in both cases, coming from Cartesian maps in (which are the same). We must show that there is a unique lift of to a chart in if and only if there is a unique lift to a chart in . Clearly it suffices to prove the "only if" implication, so suppose is a prefibration. By passing to the equalizer as in the proof that is a prefibration, we may assume that any such lift is represented by a diagram where the outer square and the triangle commutes, and is a section. Take such a diagram and let be the map representing a chart. Then the claim is there exists some zig-zag of chart equivalences identifying with . But clearly this is invariant under passing to the fiberwise opposite, and so is also a prefibration. Eigil Fjeldgren Rischel 2025 4 6 https://erischel.com/efr-ANBC/ efr-ANBC /efr-ANBC/ Stochastic charts and lenses Definition We refer to as the category of stochastic charts in . We refer to as stochastic lenses and denote it also Eigil Fjeldgren Rischel 2025 3 28 https://erischel.com/efr-IAFO/ efr-IAFO /efr-IAFO/ Example is a Markov fibration. We have already seen that it is a Markov prefibration, and that the map from the coreflection is full. So it suffices to prove faithfulness. Consider a map in , given by a diagram We can factor the section as , where the first map is just the pairing and the second is a conditional distribution. This induces a factorization of the lift over . By composing the map with this factorization to build the map , we have found a new representative for the same map. Hence every map over has a representative where the residual is . We would like to argue that, since the map is given by the conditional distribution of the composite map it is uniquely determined by it, and thus if two distinct maps in have the same underlying map in they must have equal representatives of this form, and so be identified in the coreflection (which must therefore be isomorphic to ). But of course, the two maps may only be almost certainly equal. In this case, there is a simple fix: instead of taking as the residual, take the subset given by those pairs where has positive probability given . The pairing factors over this, of course, and two maps which give the same distribution really must have the same value on every point. This proves that is faithful and hence an isomorphism. Eigil Fjeldgren Rischel 2025 4 6 https://erischel.com/efr-C5RE/ efr-C5RE /efr-C5RE/ Example , as we have noted, is a Markov prefibration, and hence induces a stochastic module structure on . This structure does not present a Markov fibration. To see this, note that in that case its fiberwise opposite would also present a Markov fibration. Then this fibration, , would be a prefibration whose deterministic part was . But Example shows that this is impossible. Eigil Fjeldgren Rischel 2025 4 6 https://erischel.com/efr-8SXB/ efr-8SXB /efr-8SXB/ Proposition Let be a Markov category with supports. Then the stochastic module induced by presents a Markov fibration. If has conditionals, this Markov fibration is isomorphic to . Eigil Fjeldgren Rischel202546Proof Given a section and , simply take the pullback to the support of . It must be the case that -almost surely, which implies strict equality on the support. Hence by Proposition , presents a Markov fibration. There is an induced map , which we claim is an isomorphism. So consider a map in : This map is in the image of where the map is taken to be a conditional. Note that every map in can be represented in this form, by taking a conditional of given to build a section to . Since conditionals are almost-surely equal, by restricting to the support of , we can find a representative which only depends on the overall map which proves that the map from is faithful, concluding the proof. Continuing from Example , we have the following trivial case: Eigil Fjeldgren Rischel 2025 4 8 https://erischel.com/efr-Z75A/ efr-Z75A /efr-Z75A/ Proposition If is Cartesian (even if it does not admit pullbacks), the definition of Markov fibration over still makes sense, the Markov fibrations are exactly the Grothendieck fibrations, and their fiberwise opposites are simply their fiberwise opposites in the usual sense. Eigil Fjeldgren Rischel202548Proof This is trivial because is simply the identity functor, hence it is monadic (with the identity monad,) hence every fibration/prefibration presents a Markov fibration, namely itself, and is in particular a Markov fibration. The fiberwise opposite is simply given by applying the identity (taking the stochastic module on the deterministic part), taking the fiberwise opposite, then applying the identity again (passing to the presented markov fibration). It is worth noting that, even in the case where is not a prefibration, it may still deserve the name "stochastic lenses". For example the stochastic lenses in can be seen to contain as a full subcategory, even though it does not form a Markov fibration (see Theorem below). Part of the motivation for the theory of dependent optics is to identify a category of stochastic optics which admits all coproducts. If is distributive, satisfies but this coproduct fails to exist in general if the two secondary objects are distinct. The idea is that this coproduct should exist as a family indexed by , where for , and for . Our theory accommodates this example under the mild additional hypothesis of extensiveness Eigil Fjeldgren Rischel 2025 4 6 https://erischel.com/efr-B2P7/ efr-B2P7 /efr-B2P7/ Definition A Markov category is said to be an extensive Markov category if it admits finite coproducts, whose injections are deterministic, and which satisfy the following equivalent conditions: If we let refer to the full subcategory of the slice spanned by the deterministic morphism , we have an equivalence of categories , given by taking coproducts is an extensive category in the usual sense and the inclusion preserves pullbacks along coproduct inclusions. Eigil Fjeldgren Rischel 2025 4 6 https://erischel.com/efr-QAV2/ efr-QAV2 /efr-QAV2/ Proposition Let be an extensive Markov category, let be a fibration which satisfies . Note that this implies admits finite coproducts, and they're given exactly by this pairing. Suppose is equipped with a stochastic module structure. Then preserves the finite coproducts. In particular, has the same coproducts as and preserves them as well. Eigil Fjeldgren Rischel202546Proof This is straightforward to check---the residual splits into by extensivity of , which also implies the section must split as the copairing of a section . By the condition on the fibration, the map in the fiber over splits into a map over and a map over . Using the extensivity again, it is straightforward to see that this decomposition respects the equivalence relation. In particular, both admit coproducts given simply as coproducts in , if is extensive. Eigil Fjeldgren Rischel 2025 3 29 https://erischel.com/efr-XFAO/ efr-XFAO /efr-XFAO/ Proposition Suppose is a Markov fibration so that each pullback functor for can be taken to be bijective on objects, and that these can furthermore be chosen strictly functorial (so that ). Then, writing objects as where is the unique object in which pulls back to under the deletion (note that this means ) we may characterize the fiberwise dual as having hom-sets Eigil Fjeldgren Rischel2025329Proof The correspondence in both directions is obvious by just formally reversing the direction of the map in a representing tuple ---the fact that this assignment respects the equivalence relation follows from the fact that the monad preserves fiberwise opposites. Eigil Fjeldgren Rischel 2025 4 8 https://erischel.com/efr-HWAZ/ efr-HWAZ /efr-HWAZ/ Limits of Markov fibrations The class of fibrations is stable under pullback. This turns into an indexed category, which represents a fibration over ---this is the "global" category of fibrations , whose objects are fibrations and whose morphisms are commutative squares where the top map preserves Cartesian morphisms. By considering universal constructions like limits and colimits in , additional fibrations can be constructed. It would similarly be useful to study limits in the category of Markov fibrations. Moreover, the products in allow one to express notions of internal pseudomonoid---these turn out to be monoidal fibrations, and this is a key part of Moeller and Vasilakopoulou's treatment of the monoidal Grothendieck construction, Reference . Since we want to study monoidal Markov fibrations, we should study their limits. Eigil Fjeldgren Rischel 2025 4 9 https://erischel.com/efr-9VAH/ efr-9VAH /efr-9VAH/ Lemma Let be a Markov prefibration, and let be any functor from another Markov category which preserves deterministic maps. Then the pullback is again a Markov prefibration, and the functor preserves Cartesian maps. Eigil Fjeldgren Rischel202549Proof Since pullbacks compose, is the pullback of along . Since fibrations are stable under pullback, this is a fibration. Now consider a triangle in : with deterministic, and let be Cartesian maps lying over these. We must show there is a unique lift rendering the lifted triangle commutative. By definition, to give such a morphism is to given one over , and the triangle commutes if and only if its image in commutes. The triangle in goes to a triangle of the same class in and the Cartesian maps go to Cartesian maps of the same type. Hence there is a unique lift of of the given type, which is exactly what we needed to show. Eigil Fjeldgren Rischel 2025 4 9 https://erischel.com/efr-Z8X4/ efr-Z8X4 /efr-Z8X4/ Remark Given any functor between Markov categories, we may attempt to define an oplax monoidal structure by pairing the projections. This is not necessarily a natural transformation. However, if it is, it automatically equips with the structure of an oplax monoidal functor. Recall that oplax monoidal functors carry comonoids to comonoids. An oplax monoidal functor between Markov categories preserves the given comonoids if and only if it is induced like this. Hence, there is at most one way to equip a functor between Markov categories with such a structure---it is a property, not extra structure. Call such a functor an oplax Markov functor. Note that oplax Markov functors preserve deterministic morphisms. (Fritz Reference defines a Markov functor to be a strong monoidal functor which preserves the comonoids. Clearly this is a proper subset of our oplax markov functors.) Eigil Fjeldgren Rischel 2025 4 9 https://erischel.com/efr-FNYQ/ efr-FNYQ /efr-FNYQ/ Definition We will denote by the category whose objects are Markov prefibrations , and whose functors are commutative squares where preserves Cartesian maps, and is an oplax Markov functor. We will let denote the category of Markov categories and oplax Markov functors. Note that by Lemma , the forgetful functor is a fibration. There is an obvious functor which carries a prefibration to the pair of its Markov category and its underlying fibration onto the deterministic part. On each fiber, this admits a left adjoint, as constructed in § . By abstract nonsense these left adjoints commute laxly with the pullbacks---that is, given an oplax Markov functor and a map of fibrations over the deterministic part, there is an induced functor over , although this assignment does not preserve Cartesian squares. However this does give a functor , left adjoint to the restriction. The category of algebras over this monad is fibred over with each fiber being the category of stochastic modules over that markov category, and we get a global functor from . In the same way, we get a global functor to which carries each stochastic module to the functor We would like to study the limits in here. At this point, we are forced to consider for a moment a bit of higher category theory. Structures on a category defined "up to isomorphism" generally don't play well together with limits in the category , since they are defined "up to equality". For example, we cannot infer from the fact that have products and the functors preserve them that the equalizer of has products, since given in the equalizer, we have , but not necessarily equality! For the moment we will restrict ourselves to limits of strict (and in particular, strong) monoidal Markov functors, since these always exist. In general one should probably consider some form of homotopy limit, but we will not go into that now. We clearly have: Eigil Fjeldgren Rischel 2025 4 9 https://erischel.com/efr-OMMA/ efr-OMMA /efr-OMMA/ Proposition Let denote the subcategory of strictly monoidal Markov functors (i.e those where the oplaxator is the identity). and admit all products, computed simply as products in . admits all finite limits, and these are preserved by the inclusion into Eigil Fjeldgren Rischel 2025 4 11 https://erischel.com/efr-G0AN/ efr-G0AN /efr-G0AN/ Proposition The category admits all products, and pullbacks along isofibrations. These are simply computed as limits in Eigil Fjeldgren Rischel2025411Proof It is immediately apparent that products (that is, pullbacks over ) of prefibrations are again prefibrations, since the Cartesian lifts can simply be computed coordinatewise, and the uniqueness property checked coordinatewise. Let be a pullback of Markov prefibrations, with an isofibration. (Note that their pullback in is simply their pullback in ). First note that since limits commute, the deterministic part is given by . Thus to prove this is a fibration, it suffices to note that fibrations are stable under pullback along isofibrations. Given and two lifts which are identified in we get two Cartesian lifts . These go to two Cartesian lifts in , and are therefore identified up to isomorphism, but we can lift this isomorphism to and obtain a pair of lifts in the strict pullback---this is a Cartesian lift. Since Cartesian lifts are given by pointwise Cartesian lifts, given a triangle and a stochastic lift we have a unique lift in both . These both go to lifts in ---since such a lift is also unique, they are identified. Hence there is a unique lift in the pullback. Eigil Fjeldgren Rischel 2025 4 10 https://erischel.com/efr-FXBP/ efr-FXBP /efr-FXBP/ Proposition The functor preserves products. For each Markov category with weak conditionals, preserves the terminal object, and pullbacks along isofibrations. Eigil Fjeldgren Rischel2025410Proof The product-preservation is clear from the description of . Let admit weak conditionals. It suffices to show that preserves the terminal object and pullbacks in The terminal fibration is . Clearly the terminal Markov prefibration is so we must show that is an isomorphism. Its morphisms are simply spans with the left leg equipped with a stochastic section , which goes to the composite . As noted before, this is clearly full, by taking , and faithful because the pairing exhibits the equality of this canonical representative with any other. Now let be a cospan of fibrations over , with an isofibration, and consider the pullback . There is a natural transformation which we must show to be an isomorphism. Maps on the left-hand side are given by a span , a stochastic section , and a map in the pullback of the fibers . A map on the right-hand side is given by two spans each equipped with a map, so that they become identified in . Let the apexes of the two spans be . It suffices to consider the case of a span with a common lifting , so that the pullbacks of the two maps to agree. But then the original maps may also be pulled back to have as the underlying span, and thus are in the completion of the pullback. Similarly, given two maps which become identified in the image, we can again use the identifying maps in to identify the original maps, proving faithfulness. This finishes the proof. Eigil Fjeldgren Rischel 2025 4 29 https://erischel.com/efr-TZ9L/ efr-TZ9L /efr-TZ9L/ Proposition For any (pullback-positive) , the monad on preserves products. Eigil Fjeldgren Rischel2025429Proof There is a canonical map . Clearly the deterministic parts are both isomorphic to , and so on this part it is an isomorphism---in particular, bijective on objects. To see it is full, consider an morphism in the codomain, given by a pair of maps , sections and maps in . Then this pair is equivalent to equipped with the pairing and the pullbacks of , which is in the image. Given two maps witnessing equations with another pair of maps, it's easy to see that this lifts to a map witnessing the identity between these, so it's faithful. This concludes the proof. Eigil Fjeldgren Rischel 2025 4 11 https://erischel.com/efr-UF7B/ efr-UF7B /efr-UF7B/ Lemma The property of having weak supports is stable under equalizers in stochastic module fibrations over . If has weak conditionals, it is also stable under finite products (hence all finite limits). Eigil Fjeldgren Rischel2025411Proof (Note that stochastic modules themselves do not admit all finite limits, requiring some sort of isofibration property---we merely claim here that if the limit exists, it again admits weak supports) It is clear that the terminal object has weak supports (regardless of ). Given an equalizer , if is a deterministic map with a stochastic section and is a map in which goes to the identity in , find a factorization so that the image in pulls back to the identity over . Then clearly the same is true for itself. Now consider a product . The point is that given a pair of maps that go to the identity, we can find where the pullbacks are the identity. We form the pullback and use the weak conditionals to find a common lift of the two given sections to this. This gives the required weak supports. Eigil Fjeldgren Rischel 2025 4 26 https://erischel.com/efr-LTEL/ efr-LTEL /efr-LTEL/ Monoidal Markov Fibrations Recall that if is symmetric monoidal, inherits a symmetric monoidal structure. At the same time, if is a monoidal fibration, the fiberwise opposite retains a monoidal structure. Since these monoidal structures play an important role both in compositional game theory (where it would not be much of an exaggeration to say the entire point is to use string diagrammatic syntax to work with games) and in categorical systems theory, it is clearly important to understand the monoidal structure on Markov fibrations. Luckily, as we will see in this section, there are essentially no difficulties in accounting for the monoidal structure. The theory of monoidal fibrations has been developed by Moeller and Vasilakopoulou, Reference , and Shulman Reference . We briefly sketch it here for convenience. There are essentially two available notions of monoidal fibration: For any category , the 2-category admits products, and we can ask for an internal pseudomonoid in this 2-category. This is equivalent to asking for a functor ---in other words, for a monoidal structure on each fiber so that the base-change functors become (strong) monoidal. The global category of fibrations admits products, and we may ask for an internal pseudomonoid here. This is what Shulman calls a monoidal fibration: a fibration where both come equipped with monoidal structures, the fibration is a strict monoidal functor, and Cartesian maps are stable under monoidal product. By a result of Moeller and Vasilakopoulou, these notions coincide in the case where is Cartesian monoidal. Since we are only interested in ordinary fibrations over which is indeed Cartesian, we may apply this result. However, our notion of monoidal Markov fibration will be a modified version of the latter. Eigil Fjeldgren Rischel 2025 4 25 https://erischel.com/efr-CSLQ/ efr-CSLQ /efr-CSLQ/ Monoidal markov prefibration Definition A monoidal Markov prefibration is a markov prefibration equipped with a monoidal category structure on so that is strict monoidal and so that the underlying fibration is a monoidal fibration (i.e so that preserves Cartesian lifts). A braided or symmetric monoidal Markov prefibration is a monoidal prefibration equipped with a braiding or symmetry on so that is moreover a braided monoidal functor. Note that a monoidal Markov prefibration is the same thing as an internal pseudomonoid in the global category of prefibrations. We will not delve further into this point, however. We will need this lemma: Eigil Fjeldgren Rischel 2025 4 26 https://erischel.com/efr-7GH5/ efr-7GH5 /efr-7GH5/ Lemma Let , be a reflexive pair of identity-on-objects, strict monoidal functors. Let be the coequalizer in . Then inherits a monoidal structure making strict monoidal. Moreover, if is symmetric or braided, inherits this structure making a braided functor. Eigil Fjeldgren Rischel2025426Proof The only thing to check is that the equivalence relation on morphisms is stable under tensoring. But this is clear: let . Then witnesses the identification of and . Tensoring on the right is analogous. This finishes the proof for the monoidal structure. In the braided or symmetric case, it is clear that the image of the braiding of in becomes a braiding on it clearly satisfies the coherence equations (being a quotient), and naturality follows by simply choosing representatives and noting that the tensor in is defined by tensoring representatives in . Finally if is symmetric clearly the equation passes to . Eigil Fjeldgren Rischel 2025 4 25 https://erischel.com/efr-85DX/ efr-85DX /efr-85DX/ Theorem The free prefibration monad on has a canonical lifting to Given a monoidal prefibration, its underlying stochastic module acquires the structure of an algebra of this lifted monad. Given an algebra for the lifted monad , acquires a monoidal structure so that is a strict monoidal functor. If moreover has weak supports, this forgetful functor is a monoidal prefibration. Every statement holds also for braided or symmetric fibrations. Eigil Fjeldgren Rischel2025425Proof By Reference , is equivalent to the category of pseudomonoids in . Since the monad preserves products, it must preserve pseudomonoids, which is all we need. Now suppose is a monoidal prefibration. Then its underlying fibration is a monoidal fibration, hence an object of . The claim is that the functor is monoidal. The induced monoidal structure is given on objects by and takes a pair of morphisms in the fiber represented by sections and to . Recalling that the algebra structure is defined by taking to the composite and chasing the below diagram around, it is apparent that the algebra structure preserves the monoidal structure. Now let be a monoidal stochastic module in this sense. It suffices to show that the free prefibration is a monoidal prefibration, by Lemma , and the induced functor will clearly be strict monoidal if is. To construct this monoidal structure on , simply note that since preserves global limits as well, there is an induced monoidal structure on so that the forgetful functor is strict monoidal. Recalling that the Cartesian lifts of to are given by the span and the morphism , it is easy to see by unwinding the definition that these are stable under tensor. If has weak supports, we have already proven that is a strict monoidal functor, and weak supports are equivalent to the claim that it is a prefibration. Since the Cartesian lifts are just the equivalence classes of the Cartesian lifts in the preceding claim that they are stable under tensor implies the same for finishing the proof. The last point is mostly trivial. The product preservation still establishes the lifting to and . Given a braided or symmetric monoidal prefibration, the braidings are Cartesian and in the deterministic part, so the underlying fibration is braided/symmetric and they are preserves by the stochastic module structure. The braiding/symmetry on follows again from Lemma , and there is nothing to show for the last point. Eigil Fjeldgren Rischel 2025 4 26 https://erischel.com/efr-T9F4/ efr-T9F4 /efr-T9F4/ Monoidal Markov fibration Definition A monoidal Markov fibration is a monoidal prefibration which is a Markov fibration. It is braided or symmetric if it is braided or symmetric as a prefibration Eigil Fjeldgren Rischel 2025 4 26 https://erischel.com/efr-KMX6/ efr-KMX6 /efr-KMX6/ Remark Since preserves both global products and fiberwise ones, it induces both a fiberwise monoidal structure and a "global" monoidal structure on . Here we only use the global one. If were Cartesian, the global one would be induced from the local one by, given maps pulling each of them back along the square (and the analogous one for ) and tensoring them over . In a Markov prefibration, of course, these pullbacks are not unique unless is deterministic, but there are "canonical" lifts given by tensoring globally with the (fiberwise) unit map over , and the global tensor is indeed given by the tensor of these canonical lifts (this doesn't provide a noncircular definition of the global tensor, of course). This provides a consistency relation between the two tensor products. Again, we will not dwell on this point. Eigil Fjeldgren Rischel 2025 4 30 https://erischel.com/efr-P4T0/ efr-P4T0 /efr-P4T0/ Markov structure on stochastic charts Proposition Let be a monoidal stochastic module fibration over and suppose each fiber is a Markov category, and this structure is preserved by the pullback functors . Then carries the structure of a Markov category, so that is a Markov functor. If is a Markov prefibration which is also a strict Markov functor, the induced monoidal structure on acquires a fiberwise Markov structure. Eigil Fjeldgren Rischel2025430Proof Every equation in the definition of Markov category involves only deterministic maps, so this can be verified entirely over . Thus this reduces to the claim: given a monoidal fibration over a Cartesian base, if each fiber has a Markov structure, the global monoidal structure has one as well. Given an object a map is by definition a map plus a map . Taking to be the copy map, the codomain there is by definition the monoidal product in the fiber and so we simply use the copying map of the fiberwise monoidal structure. Given a Markov structure on the total category we simply apply this idea in reverse and take the map to be the copy map. The deletion maps can be handled in an analogous way. Eigil Fjeldgren Rischel 2025 4 30 https://erischel.com/efr-FT8J/ efr-FT8J /efr-FT8J/ Corollary Let be a stochastic module fibration, and suppose each fiber has coproducts, and these are preserved by the pullback functors. Then is a Markov category, with monoidal structure given by The Markov structure of Corollary is a generalization of the fact that, if has products and coproducts, and the products distribute over the coproducts, then has products given by . See eg Reference , section 8 for more on this. Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-IMLW/ efr-IMLW /efr-IMLW/ Examples We have already noted several examples throughout. We'll gather a few more here, and also collect a few scattered throughout to make the picture more clear. First, let us make explicit the example of optics, as strongly as it can be stated Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-A6YR/ efr-A6YR /efr-A6YR/ Proposition Let act on . Then has a functor to . The deterministic part admits the structure of a stochastic module fibration. There is an isomorphism . If is itself symmetric monoidal and the action is symmetric (meaning it is given by for some symmetric monoidal functor , see Reference 5.4.3 and 5.5.12), this stochastic module is symmetric monoidal and the isomorphism is an isomorphism of symmetric monoidal categories. Eigil Fjeldgren Rischel2025428Proof We have essentially already seen that the deterministic part is a fibration, with maps over given by , and with the pullback functors acting by reparametrization. Since a map with deterministic marginal on is always equal to the independent pairing of we can slide the former through and identify each optic over a given with a map and note that this map is conversely an invariant of an optic, since it is obtained by postcomposing with . Given , objects and a map over classified by a section act by reparametrization. It is clear that this gives the structure of a stochastic module. The functor from optics takes to the span equipped with the obvious map that simply forgets . Note that every chart is equivalent to one of this form (given and the map exhibits the required equivalence), hence the functor is full. Moreover, note that each chart is associated with a well-defined optic, given by the maps . Is is straightforward to see both of these maps are preserved by chart equivalence. This gives an inverse to the functor, proving it is faithful. Since it is identity on objects, this finishes the argument. In the symmetric monoidal case, it is immediately clear that the fibration on is symmetric monoidal. Since the action is symmetric monoidal, given maps and it is clear that composing to get maps then tensoring and composing with the diagonal to get gives the same map as tensoring, then using the map . Hence we have a symmetric monoidal module. It's straightforward to see the functor is symmetric monoidal, and that finishes the argument. Slightly orthogonally, we have the following comparison between and : Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-K6NM/ efr-K6NM /efr-K6NM/ Theorem Let be any pullback-positive Markov category. Then is a Markov prefibration which thus induces a stochastic module structure on . Writing simply for , we have: There is a functor which is fully faithful. Dually there is a functor which is fully faithful. and both admit symmetric monoidal structures, which make the functors strict symmetric monoidal, as well as the functors strong symmetric monoidal. If is extensive, this functor preserves the coproducts and both admit all finite coproducts. If moreover has conditionals and supports, Thus we have our previous claim that contains . We also have some examples of an analytic flavor: Eigil Fjeldgren Rischel 2025 4 28 https://erischel.com/efr-TO9K/ efr-TO9K /efr-TO9K/ Example Given a compact Hausdorff space , a Banach space bundle is a space over , , equipped with a fiberwise (complex) vector space structure , , so that there exists a cover of so that for each , there exists a Banach space and a homeomorphism over which is moreover linear in each fiber, where is equipped with the norm topology. Note that this determines a local norm on each (and in particular each ) up to equivalence (but no stricter than that). In particular each is a Banach space. A morphism of Banach space bundles is a continuous map over which is linear on each bundle. Note that this implies that on a suitable cover the maps obey for some , for any local norms inducing the topologies, and hence by compactness there exists some so that for each . If is a continuous map, there is a pullback functor . The Grothendieck construction of this gives a fibration Note that Tychonoff spaces include all compact Hausdorff spaces. Therfore consider the full subcategory spanned by these. The fibration admits the structure of a stochastic module: given , a kernel, and a linear continuous map given there is an induced function given by . Since this is bounded (being continuous on a compact space) it is (Bochner) integrable, define to be this integral. This example is analogous to optics for the action of categories of markov kernels on categories of vector spaces (as in Proposition ). Note that we do not expect this type of example to present a Markov fibration. The reason is simply that, given a parametrized linear map and a measure on , the fact that the expectation map is the identity by no means implies that the original map is almost surely the identity or anything like that. If and has positive probability, this can be canceled out by . This is impossible for probability kernels. If we add an assumption of positivity, it seems plausible that examples of this type will present Markov fibrations---but of course, that brings us quite close to categories of Markov kernels in any case. Note that, as in this example, we do not generally expect to yield a Markov fibration if is not another Markov category (and not even then in general, as the case of shows).---for these, we expect to need a sort of positivity in the fiber as well, which restricts us to things that look like probability kernels. Eigil Fjeldgren Rischel 2025 5 2 https://erischel.com/efr-O6GQ/ efr-O6GQ /efr-O6GQ/ Stochastic module of -algebras Proposition Let be a representable, positive Markov category so that admits intersections and the probability monad preserves them. Then each slice inherits a monad structure given by . This is pseudofunctorial in . Moreover, the stochastic module structure on extends to a stochastic module structure on the fibration representing the pseudofunctor . Eigil Fjeldgren Rischel202552Proof The monad is induced by the adjunction . Let (deterministic). For abstract reasons there is a natural transformation . Writing this out, we find By representability, the unit is a monomorphism. Hence a map into is precisely a map in into so that the marginal on is deterministic. But by pullback-positivity this is precisely a map (in ) into . Let , two -algebras , and a map which is a homomorphism for the induced -algebras, and a map be given. Note that by the above, . Then the induced map is given by We must show this is a -homomorphism. Let us simplify by working internally to ---thus we have a Cartesian category equipped with a strong commutative monad , a map , and a map which is a parametrized algebra homomorphism, in the sense that the diagram commutes. Now we must show the map is a -homomorphism. Write for the structure maps of the two algebras. Consider this diagram: The triangle at the top left commutes because is strong. The square to the right does not commute in general---however, since is commutative, the composite maps agree. Since the map factors over this, we may replace one edge of this square with another. The square to the right of that is simply applied to the previous diagram, and so commutes by assumption. The "triangle" under that is just two copies of the same maps, so commutes. The square on the left of the diagram commutes by functoriality of product. The square to the right of that commutes again because is strong. Hence the outer square commutes, which is precisely the homomorphism property we wanted. It is apparent that, if are free algebras, this restricts to the stochastic module structure of (viewing as the Kleisli category of ). But since every algebra is a coequalizer of free algebras, it follows that the action on general algebras is determined uniquely by this. This implies the equations of a stochastic module. As in Example , this cannot be expected to come from a Markov prefibration in general. The vast majority of examples seem to occur as subcategories of stochastic modules of the form given by Proposition (of course, is just the subcategory spanned fiberwise by the free algebras). In fact, since a stochastic module necessitates in some sense an action of on the objects of the fiber, it seems they do all have this form in a generalized way, although we have not found a better way to make this precise than the existing definition of stochastic module. Eigil Fjeldgren Rischel 2024 6 13 https://erischel.com/efr-000D/ efr-000D /efr-000D/ The construction in generic 2-categories Eigil Fjeldgren Rischel 2024 7 22 https://erischel.com/efr-003P/ efr-003P /efr-003P/ Introduction Myers' theory of categorical systems theory (see § ) gives a rich categorical structure to a wide variety of types of dynamical system. The central idea can be summarized by saying that there are two different, but tightly related, notions of morphism at play in dynamical systems. Open dynamical systems themselves involve a bidirectional information flow, captured by the notion of lens and composition of such lenses describes the composition of subsystems into systems. But morphisms between systems are unidirectional, captured by the notion of chart . Algebraically, the relationship between these two notions is that they assemble into a double category, which indexes the category of systems. There is another double category involving lenses which has been considered in the categorical study of systems. That is the double category of parametrized morphisms, applied to the monoidal category of lenses. These have been studied as an abstraction for gradient descent in several papers, see eg Reference , Reference , (the idea goes back to Reference ). Given any action of a monoidal category on another category (most simply, if is monoidal it acts on itself,) we obtain a category of parametrized morphisms (where ,) and these turn out to be extremely useful. We will mention two applications: A parametrized lens is essentially what is called a learner in Reference ---it contains the information necessary to compute a new parameter value given an existing and a sample pair , as well as the additional information required to compose such things. Thus the functoriality of backpropagation can be derived from two facts: the reverse derivative defines a monoidal functor and the construction taking a category to its category of parametrized maps, is itself functorial. This viewpoint has been significantly developed in Reference and other papers. An open game, in the sense of Hedges Reference , is almost the same thing as a parametrized lens , equipped with a subset . Given a context for the game---that is, a state (describing the state of information when the decision is made) and a continuation (describing how decisions map to outcomes ,) we obtain a function , and we say is an equilibrium strategy if . Since and this neatly captures the extra data of an open game in terms of the category of lenses. The potential of this idea as a generalized approach to "cybernetic systems" is explored in Reference . In this chapter, we will develop the theory of the category of parametrized maps. We will begin by reviewing the existing literature briefly. We will describe a double categorical version of this category---this does not seem to have appeared in the literature yet, although it has been folklore for at least a few years (and there is nothing complicated about this construction, certainly). The remainder of this chapter will be dedicated to lifting this construction to a generic 2-category (with the above being the specialization to ). We will derive this lifting using the machinery of 2-category theory. We will see how this generalization accounts for much structure which can be seen to exist on , such as its symmetric monoidal structure (assuming are symmetric monoidal). But the true application of this will be in § , where we use this to construct a triple category of open dynamical systems. The definition of the double category involves two types of categorical structure with which the reader may be unfamiliar---actegories, which are the input to the construction, and pseudo double categories, which are the output. Since we will shortly introduce the abstract internal versions of these, internal pseudomonoid actions and internal pseudocategories, we will not give a separate introduction here. The reader who is unfamiliar with these should refer to Reference for actegories, and Reference for (pseudo) double categories. A reader who simply needs a definition may look at Definition and Example . Eigil Fjeldgren Rischel 2024 7 2 https://erischel.com/efr-002H/ efr-002H /efr-002H/ The Para Construction as a double category In many different situations, we want to understand some morphism as parametrized by some data. For example, in machine learning one tries to find a function with some desirable behaviour by choosing a parametrization and searching for some so that has this behaviour (for example by gradient descent on ). In situations where we want to understand as being built up as a composite of multiple functions (for example, the layers of a neural network), it is convenient to introduce a category of parametrized morphisms, where the composition combines the parameter spaces of each composite map. We can do this in a general setting with the following definition: Eigil Fjeldgren Rischel 2024 7 2 https://erischel.com/efr-002G/ efr-002G /efr-002G/ Definition Let be a monoidal category, and let be an action of on another category . Then the construction is the category where Objects are objects of Morphisms are tuples up to the natural notion of isomorphism. Composition is by tensoring the parameter objects and composing. The fact that the parametrization object may live in a different category than the domain and codomain objects, which initially seems like a superfluous generalization, is in fact highly useful. For example, we will often want to consider parametrized morphisms where the parameter space is decorated with some additional data, for example a probability distribution. In most such cases, the category of spaces decorated with such data will act on the category of spaces without such data (simply by forgetting the data and tensoring), and hence we can realize such parametrized morphisms as examples of this construction. We may also want only Euclidean parameter space (so that we can run gradient descent simply), but allow the parametrized morphism to go between more general manifolds. , and generalizations of it, have been introduced many times. One of the earliest occurrences is by Hermida and Tennent, Reference , in the special case of a symmetric monoidal category acting on another such via a functor (by a result of Capucci and Gavranović, Reference , every "structure-preserving" action of a symmetric monoidal category on another has this form). Hermida and Tennent actually give a very intuitive universal property of : it is freely generated by adding a morphism for each , subject to the equations that this must be a monoidal natural transformation. This type of idea actually goes all the way back to Pavlović in Reference . The notation was introduced inReference , where the special case of parametrized morphisms of euclidean spaces was used to study gradient descent. In Reference , a bicategorical variant (replacing the quotient by isomorphism in the above variant in the obvious way) is introduced. Eigil Fjeldgren Rischel 2025 4 29 https://erischel.com/efr-MN7C/ efr-MN7C /efr-MN7C/ Example Let be a monoidal category, and let be any functor. Recall that is the category whose objects are pairs and whose morphisms are so that . If is lax monoidal (for ,) acquires a lax monoidal structure making the forgetful functor strong (even strict) monoidal. With the action induced by this functor, the bicategory has morphisms given by maps and maps given by reparametrizations which preserve the decoration . For example, let be a Markov category and let . Then morphisms are parametrized maps equipped with a measure on the parameter space. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-WEST/ efr-WEST /efr-WEST/ Example If regarded as a discrete 2-category, as noted, a pseudomonoid action is just a monoid action in the ordinary sense---that is, a monoid , a set and a function so that . The para construction is then the action category, whose objects are the points of and whose morphisms are elements so that (with multiplication as composition). Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-QH7O/ efr-QH7O /efr-QH7O/ Example Let be the 2-category of -enriched categories, functors and natural transformations. Recall that a ring is the same as a one-object category enriched in , and it admits a monoidal structure if and only if it is commutative (this is not particular to -enriched categories). An enriched -action on an -category is then an -module structure on each hom-set so that composition is -linear in each variable. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-KL7V/ efr-KL7V /efr-KL7V/ If a category has coproducts, then acts on it via . (This is the -enriched case of what in enriched categories is called a copower or tensor, the dual of the power objects from Example ). The morphisms of are pairs , which compose in the obvious way. Note that this definition clearly makes sense even if does not actually have coproducts. This is an example of another construction which has been called , which takes a monoidal category and a -enriched category and constructs a double category where the morphisms are pairs . It is not hard to see that this also extends to a double category in the same way, but we do not presently know the correct definition of internal enriched object that would replace pseudomonoid actions to replicate our general theory for this case. (note that the literature contains a notion of internal enriched category, Reference , but these are categories enriched over internal monoidal categories---that is, the ambient category is a 1-category and the categorical structure of the base of enrichment is formulated on top of this, not as part of the structure of the objects of the category ) There is a very natural double categorical version of , where the vertical morphisms are just the unparametrized morphisms of , and the 2-cells with this boundary: =4pt, Rightarrow, from=0, to=1, "g"] \end {tikzcd} ]]> are morphisms (if is parametrized by , by ) so that the square commutes. This notion is not original---we learned of it from Matteo Capucci---but it seems to have remained somewhat on the level of folklore. Our main goal for this section will be to construct this (pseudo) double category, and show that it is functorial---that is, given a homomorphism of actions, there is an induced functor between double categories. In fact, we will show that this construction works in any 2-category. Later we will apply it to a pseudomonoid action in (strict) double categories to construct a pseudocategory internal to double categories---our triple category of bimachines. Eigil Fjeldgren Rischel 2025 4 29 https://erischel.com/efr-55CC/ efr-55CC /efr-55CC/ Theorem Let be a monoidal category, and let be a category with an -action. Then there is a pseudo double category with , with domain and codomain given by the two projections to The identities map is given by where is the left unitor of . The horizontal composition map is the composition in given their composite is given by The horizontal composition of 2-cells is defined analogously. Moreover, if is strict homomorphism of -modules, there is an induced pseudofunctor . If is a strict monoidal functor, then regarding as an -module along this map, there is an induced pseudofunctor . These combine into a 2-functor between the 2-category of actions and strictly linear functors and the category of pseudo double categories and strict double functors. This functor preserves (strict) limits. Eigil Fjeldgren Rischel2025429Proof Note that every actegory is equivalent to a strict action of a strict monoidal category (in the sense that there exists strong monoidal equivalence and an equivalence such that these maps together form a map of actegories). It follows that it suffices to show unitality and associativity for our composition in the strict case (since these are plainly preserved by equivalence). Let , be a strict monoidal functor, and let be a linear functor "over ", that is a (strict) -linear functor when is viewed as a -actegory along . Then the induced functor is given by on the vertical category, and on the horizontal category by where the isomorphism is the linearity---noting that the -action on is precisely acting by . By naturality of the lineator it is straightforward to see that this is a functor, and it clearly preserves domain and codomain. It preserves units (strictly!) by the compatibility between the unitor and lineator. It remains to see that this preserves composition. For brevity, denote the induced functor from here. It suffices to verify this for strict actions. So the composite of is given by the composite Given two such composable maps, call them , the two objects are given by ,where are respectively the maps across the top and bottom of this diagram: This proves that preserves the composition strictly, as desired. (The rightmost triangle commutes by definition, and the leftmost rectangle is a lineator coherence) Since the strict limits in and are computed levelwise, it suffices to observe that the slice category also preserves limits, being a weighted limit itself. Eigil Fjeldgren Rischel 2025 4 29 https://erischel.com/efr-HUEE/ efr-HUEE /efr-HUEE/ Remark Note that comes equipped with a strict functor to (viewed as a double category). In fact, one can recover the action from this data: the object is characterized up to isomorphism by its universal property: parametrized maps with parameter are in bijection with maps parametrized by . Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-9I1M/ efr-9I1M /efr-9I1M/ 2-Limit sketches Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-L9JB/ efr-L9JB /efr-L9JB/ Weighted limit Definition Let be a 2-category, let be a 2-diagram in it, and let be another 2-functor. A limit of weighted by {W} is an object equipped with a natural isomorphism of categories (natural in ). Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-K3YI/ efr-K3YI /efr-K3YI/ Particular weighted limits Example If is constant at the point, and is an 1-category, then a weighted limit is simply a in the underlying category so that there is additionally an isomorphism of categories . Note that this is stronger than just a limit in . We will refer to limits of this form by their ordinary names---speaking for example of pullbacks, products, and so on. If , , and the universal property of the weighted limit is that . In this case we write for this limit if it exists, and call it a power of by . If and the weighting is given by (with the obvious inclusions), then the weighted limit is called the comma object and will be denoted . Note that if this is precisely the ordinary comma category. The analogue for a cone on an object in the setting of weighted limits is called a cylinder: it is a natural transformation . A cylinder on induces a natural transformation and we say it's a limit cylinder if this is an isomorphism. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-40Q5/ efr-40Q5 /efr-40Q5/ 2-limit sketch Definition A 2-limit sketch is a small 2-category equipped with a (small) set of cylinders . A model of the sketch in is a 2-functor which carries each cylinder in to a limit cylinder. We write the category of models and natural transformations (leaving the set of cylinders implicit). If we write simply . Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-D7EH/ efr-D7EH /efr-D7EH/ Lemma Let be a 2-limit sketch. Then the functor given by is fully faithful, and its essential image consists of those functors so that each is representable, Eigil Fjeldgren Rischel202554Proof First note that the codomain can be identified with the subcategory of spanned by those where each is a model. Since the Yoneda embedding preserves limits, the functor from clearly lands inside here. Since is itself a full subcategory of the functor category this is fully faithful. Clearly for each model and for each the functor is representable, by . Conversely, if is such that each is representable, then the currying of factors over , and since the Yoneda embedding preserves those limits that exist, this factorization must be a model as well. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-9O7K/ efr-9O7K /efr-9O7K/ Proposition Let be limit sketches and suppose given a functor which preserves limits and is accessible, that is it preserved -filtered colimits for some . Then admits a left adjoint . In particular, Eigil Fjeldgren Rischel 2025 3 21 https://erischel.com/efr-ZRUW/ efr-ZRUW /efr-ZRUW/ Pseudocategories Given a 2-category , there is a natural loosening of the notion of "internal category in ", given by requiring that associativity and unitality hold only up to chosen coherent isomorphism 2-cells. This is the notion of pseudocategory. We refer to Reference for a thorough description of this concept (as well as a review of the earlier literature). However, we will review the basic facts here for convenience. Eigil Fjeldgren Rischel 2025 3 21 https://erischel.com/efr-ZRUX/ efr-ZRUX /efr-ZRUX/ Internal pseudocategory Definition Let be a 2-category with pullbacks. Then an internal pseudocategory in (or simply a pseudocategory in ) consists of the following data: Two objects Morphisms so that . A morphism , so that Isomorphism 2-cells , , and Satisfying the following equations: , And so that the following diagrams commute: A homomorphism or strict functor of pseudocategories is a pair of morphisms which commute with all the structure---that is, , , and so on. There is a clear notion of natural transformation of homomorphisms. We write for the 2-category of pseudocategories, homomorphisms and natural transformations in . It is important to note that, although this is a weakening of the definition of internal category, pseudocategories are themselves a strict concept---they are defined in terms of equations that must hold up to strict equality. Thus for example there is an enriched monad (a 2-monad) on whose strict algebras are the pseudo double categories. Eigil Fjeldgren Rischel 2025 3 21 https://erischel.com/efr-ZRUY/ efr-ZRUY /efr-ZRUY/ Example An internal pseudocategory in is, as mentioned above, a pseudo double category. An internal pseudocategory with terminal is the same thing as an internal pseudomonoid. If we fix a specific terminal object and require , this is an isomorphism of 2-categories (both for the categories of pseudomorphisms and homomorphisms). An internal pseudocategory with identities is the same thing as a (strict) internal category. In particular, internal pseudocategories in discrete 2-categories are merely internal categories. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-LTJQ/ efr-LTJQ /efr-LTJQ/ Example Let be the 2-category of monoidal categories, strict monoidal functors and monoidal natural transformations. Note that this has pullbacks. Then the objects of are a stricter version of monoidal pseudo double categories: they are pseudo double categories where both carry a monoidal structure so that are strict monoidal functors and are monoidal natural transformations. Here we are already beginning to feel the limitations of this internalization a bit---really we should ask for , at least, to be only a strong monoidal functor. But we move on for now with this problematic definition. Of course, there is a very crucial notion of pseudofunctor between pseudo double categories. This has an internal formulation in terms of pseudomorphisms: Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-8YZ9/ efr-8YZ9 /efr-8YZ9/ Pseudomorphism of pseudocategories Definition Let be internal pseudocategories in a 2-category . A pseudomorphism consists of the following data: Morphisms , in Isomorphism 2-cells and So that the following equations hold: And so that the following diagrams commute: One should not spend too much time looking at Definition . This is clearly a horrible concept, and we will prefer to work around it rather than referring to it explicitly. It may be a useful exercise to convince oneself that, when and the pseudocategory is a pseudomonoid---eg a monoidal category---this coincides with the notion of strong monoidal functor. One can also define lax and oplax functors between pseudo double categories, but we will not need that concept here. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-7ZZG/ efr-7ZZG /efr-7ZZG/ Proposition There exists a -limit sketch whose strict models are internal pseudocategories, and whose strict natural transformations are homomorphisms. Eigil Fjeldgren Rischel202554Proof There is nothing surprising about this to those familiar with limit sketch presentations of other algebraic gadgets, but for completeness we give an explicit construction here. Take to have objects . The 2-category is freely generated by these objects and the following data: Morphisms Morphisms for so that commutes. Morphisms and satisfying Further morphisms, and invertible 2-cells as in the following diagrams: =4pt, Rightarrow, from=1-2, to=2-1] \arrow ["m", from=1-2, to=2-2] \arrow ["m"', from=2-1, to=2-2] \end {tikzcd} ]]> =3pt, Rightarrow, from=0, to=1] \arrow ["\rho "', shorten <=3pt, shorten >=3pt, Rightarrow, from=2, to=1] \end {tikzcd} ]]> and all the other induced morphisms appearing in Definition , subject to those equations holding. Equations asserting that the morphisms and 2-cells with codomain 1]]> behave as expected under postcomposition with the pullback projections. The prescribed pullback cones are the diagrams involving for . An explicit description of this sketch seems not to have appeared in the literature until Reference (example 5.13). Note that they construct a richer object, what they term an -sketch, which includes the information that the domain and codomain are "tight" and must be preserved strictly by pseudofunctors, but composition and identities are "loose" and may be preserved only up to natural isomorphism. This idea will be highly useful later, although we will not go into a full treatment of their theory. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-NI76/ efr-NI76 /efr-NI76/ Actegories Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-OFNV/ efr-OFNV /efr-OFNV/ Actegory Definition Let be a monoidal category. A -actegory (also -module, -action) is a category equipped with a functor called the action (written infix, ), and natural isomorphisms satisfying the following coherence equations: Actegories are a very natural concept, and as one would expect their history goes back a long way. The concept has been considered by Benabou all the way back in Reference , and used many times since them. We will not delve too deeply into their theory here---see Reference for a thorough treatment. Just as pseudo double categories, actegories really make sense in every 2-category, as a weakening of monoid actions. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-IOLJ/ efr-IOLJ /efr-IOLJ/ Pseudomonoid action Definition Let be a 2-category. A pseudomonoid action internal to consists of the following data: An internal pseudomonoid , an object , a morphism natural isomorphisms and , satisfying the coherence equations from Definition . A strict homomorphism of actions is a pair so that is a strictly monoidal functor and preserves the action strictly, i.e , etc. Note that this makes sense even if the monoidal category or action is not itself strict. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-6S13/ efr-6S13 /efr-6S13/ Proposition There is a limit sketch whose models are tuples of a pseudomonoid , an object and an action of on . The strict natural transformations between models are strictly linear functors. Eigil Fjeldgren Rischel202554Proof As above, there is nothing difficult about this. Let contain as a subcategory the sketch of pseudomonoids , along with three additional objects projections , and limit cones expressing these as the products and respectively, a morphism and isomorphism 2-cells and , subject to the coherence equations in Definition . Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-I33G/ efr-I33G /efr-I33G/ Discrete pseudomonoid Example If is an ordinary category with finite products (viewed as a discrete 2-category), a pseudomonoid is simply an internal monoid, and an action is just a monoid action in the ordinary sense. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-ARIX/ efr-ARIX /efr-ARIX/ Monoidal and symmetric monoidal actegories Example In Reference , the authors study actegories with extra monoidal structure. Although they do not introduce pseudomonoid actions in a general 2-category, they study actegories internal to monoidal categories, which they call monoidal actegories. It is straightforward to check that pseudomonoid actions in agree with their notion: such an action consists of a braided monoidal category , an ordinary monoidal category , a monoidal functor and monoidal natural transformations satisfying the coherence equations for an actegory. (Note that since the functor appears on one side of , to speak of being a monoidal natural transformation we must have a monoidal structure on ---this makes into a braided monoidal category). For the case of symmetric monoidal actegories, since is cocartesian, . Given two symmetric monoidal categories Reference show that an action is simply given by a strong symmetric monoidal functor (and in this case the action is ). Unwinding this construction we see that is bi-equivalent to a category which has as objects strong (but not strict) symmetric monoidal functors and morphisms squares of symmetric monoidal functors, where the top map is strict, and which commute strictly. Again, as expected, the notion of strict linear morphism is usually too strict. One can expect an equation like to hold only up to coherent isomorphism. Let us now make this clear: Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-L5QG/ efr-L5QG /efr-L5QG/ Pseudolinear map Definition Let be a 2-category, and let be internal pseudomonoid actions. A pseudolinear map (or just linear) is a pair of maps , equipped with a pseudomonoid pseudohomomorphism structure on and a natural isomorphism satisfying the following coherence conditions (note that writing functors using element-notation like this is well-defined) In Reference , these are defined in two steps: first linear functors for the same (those with ) are considered. Then, given a strong monoidal a functorial assignment of an -action to every -action is constructed, and the full category of actions is defined as the Grothendieck construction of this. There is nothing preventing this from working in the setting of a general 2-category, and it is straightforward to verify that our notion of linear morphism of actions agrees with theirs. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-GZH4/ efr-GZH4 /efr-GZH4/ for a general 2-category We are now almost ready to prove: Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-I897/ efr-I897 /efr-I897/ Theorem Let be a 2-category which admits pullbacks, products and comma objects. Then there is a functor from the 2-category of pseudomonoid actions and pseudolinear maps to the 2-category of pseudocategories and pseudofunctors, with and . This functor preserves strict maps, limits, and filtered colimits. The main ingredient missing is a characterization of the respective notions of pseudomorphism in terms of the limit sketches. This we do now: Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-NZ0X/ efr-NZ0X /efr-NZ0X/ Proposition Let be a 2-category and let be pseudocategories, represented as models of the theory. Then a pseudonatural transformation which is strict on and the limit projections is equivalently a pseudomorphism between the pseudocategories. Now let be pseudomonoid actions. Then a pseudonatural transformation which preserves the product projections strictly is equivalently a pair of a pseudomorphism and a functor which is pseudolinear with respect to the action of on . Eigil Fjeldgren Rischel202554Proof Both of these are essentially a matter of unwinding the definitions. Let us start with pseudocategories. The requirement that is strict on the product projections amounts to requiring that is given strictly as the pullback and not merely up to 2-isomorphism. With this, is just a naturality transformation for , and is a naturality transformation for . Note that since every map in is induced from and the pullback structure, this means the other naturality transformations are determined by The composite is the naturality transformation for the map , and analogously for the composite . Hence the hexagon is a pseudonaturality coherence for the 2-cell . Analogously the two squares can be obtained as coherence 2-cells. Conversely, these suffice to make a pseudonatural transformation, again since all the other 2-cells are generated by . Now let us look at pseudomonoid actions. As above, such a pseudonatural transformation is determined by the functors and plus some 2-cells involving these and their products. As a special case of the above when we find that the restriction of the pseudonatural transformation to the pseudomonoid part is equivalently a strong monoidal functor. Now the required lineator is a naturality square for the multiplication map . As above this determines all the other naturality transformations because all the other maps are generated from (and the pseudomonoid structure) using the product structure. Also analogously to the above argument, the coherence pentagon is a pseudonaturality coherence for the natural isomorphism , and the coherence triangle for . Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-T667/ efr-T667 /efr-T667/ Remark This characterization of the notions of pseudomorphism cannot help but seem a little ad hoc. The basic idea here seems to go back to Power in Reference . After constructing an enriched Lawvere theory (which is just a special kind of limit sketch) corresponding to any finitary enriched monad, he notes that in the -enriched case the pseudomorphisms of monad algebras correspond to exactly those pseudonatural transformations which preserve the products strictly. An elegant theory which generalizes this basic idea has recently been developed by Bourke, Ko, and Varkor in Reference (as we mentioned briefly before). Their theory involves decorating the sketch with a set of tight morphisms, which the transformations must be strictly natural with respect to. Beyond developing a general theory of this (not specialized to a few examples as above,) they also show how to construct the categories of (op)lax homomorphisms, and develop a commutativity of internalization principle for their models. This could be profitably applied in our case to understand better, for example, the implied notion of monoidal triple category, but we will not pursue this here. Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-0BHI/ efr-0BHI /efr-0BHI/ Proof of Theorem Proof First recall that can be identified with the subcategory of spanned by the levelwise representable presheaves. Note that for it suffices to verify representability for since the rest are pullbacks of these, and pullbacks of representable presheaves are again representable since admits pullbacks. Postcomposition with the previously-constructed functor gives a functor . Clearly this functor preserves representability (again, since limits of representable functors are representable). This gives the desired functor . Under the identification of with a subcategory of it is clear that the pseudonatural transformations of models correspond to those pseudonatural transformations which are strict on morphisms in . This implies the full category of models and pseudonatural transformations can be identified with the subcategory of spanned by those 2-functors which come from models---that is, must factor over . (To be clear: the morphisms in are strictly natural transformations between functors , but each component of such a natural transformation is a pseudonatural transformation of models). Under this identification, it is clear that pseudofunctors correspond to those natural transformations in which are valued in pseudofunctors, and analogously for pseudolinear morphisms and maps . Hence it suffices to observe that the -valued version carries pseudolinear maps to pseudofunctors. Note that is given objectwise as a finite limit. Moreover, since the limit sketch of pseudocategories only involved finite limits, the class of pseudocategories is stable under filtered colimits in . Hence commutes with filtered colimits, and is in particular accessible. Hence it admits a left adjoint where is the restriction of the left adjoint of . Both forming the hom-category and precomposing with are 2-functors, and as such preserve pseudonatural transformations. Thus it only remains to observe that also preserves the partial strictness property, but this clear. We have not yet given any thought to functoriality in , but passing through the constructions, it is apparent that we have: Eigil Fjeldgren Rischel 2025 5 4 https://erischel.com/efr-TY4G/ efr-TY4G /efr-TY4G/ Proposition If is a 2-functor that preserves finite 2-limits, the square commutes up to strict natural isomorphism. (In particular, the square involving the subcategories of strict homomorphisms also commutes up to natural isomorphism). (It may seem wrong that, after working strictly all this time, this square commutes only up to natural isomorphism, but in fact that is the strict notion---the weak version of this statement would be that it commuted up to natural equivalence. Note that eg. the comma objects are only characterized up to isomorphism, so this is really the best we can hope for) The main problem with Theorem is that for many 2-categorical notions of interest, requiring (for example) the action to be a strict homomorphism of whatever structure under consideration is too strict to work, while working with the full category of pseudomorphisms prevents the pullbacks required for pseudocategories from existing. Thus for example, an internal pseudomonoid in is a commutative monoidal category, which is far too strict for most purposes---generally speaking, symmetric monoidal categories can not be strictified into commutative ones. In § , we will want to construct a symmetric monoidal "triple category"---that is, a symmetric monoidal pseudocategory in strict double categories. We will manage this via the preceding by noting that since preserves products, it carries (symmetric) pseudomonoids to pseudomonoids---in other words, we can apply the internalization the other way around, taking pseudomonoids in actions rather than actions in pseudomonoids. This works because strict products still exist in the category of pseudohomomorphisms, but for a more general categorical structure, we would be in trouble. Contributions