The \mathsf {Para} construction in generic 2-categories › Introduction [efr-003P]

Myers' theory of categorical systems theory (see § [efr-0023]) 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 {X' \choose X} \leftrightarrows {Y' \choose Y}, and composition of such lenses describes the composition of subsystems into systems. But morphisms between systems are unidirectional, captured by the notion of chart {X' \choose X} \rightrightarrows {Y' \choose Y},. 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 [bruno-etal-categorical-learning-2021], Reference [bruno-thesis-fundamental-components], (the idea goes back to Reference [backprop-as-functor]). Given any action of a monoidal category \mathcal {M} on another category \mathcal {C} (most simply, if \mathcal {C} is monoidal it acts on itself,) we obtain a category of parametrized morphisms f: X \cdot P \to Y (where P \in \mathcal {M}, X, Y \in \mathcal {C},) and these turn out to be extremely useful. We will mention two applications:

1. A parametrized lens {P \choose P} \otimes {X \choose X} \to {Y \choose Y} is essentially what is called a learner in Reference [backprop-as-functor]---it contains the information necessary to compute a new parameter value p' given an existing p \in P and a sample pair x \in X, y \in Y, 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 \mathsf {Euc} \to \mathsf {Lens}(\mathsf {Euc}), and the construction \mathsf {Para}(-), taking a category to its category of parametrized maps, is itself functorial. This viewpoint has been significantly developed in Reference [bruno-etal-categorical-learning-2021] and other papers.
2. An open game, in the sense of Hedges Reference [hedges-etal-comp-gametheory], is almost the same thing as a parametrized lens {\Sigma ' \choose \Sigma } \otimes {S \choose X} \leftrightarrows {R \choose Y}, equipped with a subset E \subset \Sigma \times \mathsf {Set}(\Sigma ,\Sigma '). Given a context for the game---that is, a state x \in X (describing the state of information when the decision is made) and a continuation Y \to R (describing how decisions y \in Y map to outcomes r \in R,) we obtain a function k: \Sigma \to \Sigma ', and we say \sigma \in \Sigma is an equilibrium strategy if (\sigma ,k) \in E. Since \mathsf {Set}(\Sigma ,\Sigma ') = \mathsf {Lens}(\mathsf {Set})({\Sigma ' \choose \Sigma },I), and \Sigma = \mathsf {Lens}(\mathsf {Set})(I,{\Sigma ' \choose \Sigma }), 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 [towards-cybercat].

In this chapter, we will develop the theory of the category \mathsf {Para} 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 \mathbb {C} (with the above being the specialization to \mathbb {C} = \mathsf {Cat} ). 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 \mathsf {Para}, such as its symmetric monoidal structure (assuming \mathcal {M},\mathcal {C} are symmetric monoidal). But the true application of this will be in § [efr-ZRUZ], where we use this to construct a triple category of open dynamical systems.

The definition of the double category \mathsf {\mathbb Para} 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 [actegories-amthematician-capucci-gavranovic] for actegories, and Reference [shulman-monfibs] for (pseudo) double categories. A reader who simply needs a definition may look at Definition [efr-OFNV] and Example [efr-ZRUY].