Let p: \mathcal {D} \to \mathcal {C} be a functor. Given X \in \mathcal {C}, write \mathcal {D}_X for the (strict) pullback \{x\} \times _\mathcal {C} \mathcal {D}. Explicitly, this consists of the objects in \mathcal {D} with p(A) = X and the morphisms with p(f) = 1_X.

Let f: X \to Y \in \mathcal {C} be a morphism.

1. A map \bar {f}: \bar {X} \to \bar {Y} with p(\bar {f}) = f is locally Cartesian if for each \bar {X}' with p(\bar {X}') = X, postcomposition with \bar {f} induces a bijection \{g: \bar {X}' \to \bar {X} \mid p(g) = 1_X\} \xrightarrow {\sim } \{g' : \bar {X'} \to \bar {Y} \mid p(g') = f\}
2. A map is Cartesian if for every g: Z \to X and \bar {Z} with p(\bar {Z}) = Z, there is a bijection \{\bar {g} : \bar {Z} \to \bar {X} \mid p(\bar {g}) = g\} \to \{\bar {g}' : \bar {Z} \to \bar {Y} \mid p(\bar {g'} = fg)\}, note that every Cartesian map is locally Cartesian (take g = 1_X)
3. p is a Grothendieck fibration (or just fibration) if, for every \bar {Y} \in \mathcal {D} such that p(\bar {Y}) = Y, there exists a Cartesian map \bar {f}: \bar {X} \to \bar {Y} (for some \bar {Y}) so that p(\bar {f}) = f

p: \mathcal {D} \to \mathcal {C} is a Grothendieck fibration if and only if every f admits a locally Cartesian lift, and the class of locally Cartesian morphisms in \mathcal {D} is stable under composition.

Let \mathcal {D} \to \mathcal {C} be a Grothendieck fibration. For every f: X \to Y, \bar {Y} \in \mathcal {D}_Y, select a Cartesian lift f^*\bar {Y} \to \bar {Y} of f. Then there is a unique extension of f^* to a functor \mathcal {D}_Y \to \mathcal {D}_X so that the squares

If \mathcal {D} \to \mathcal {C} 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-).

Let \mathcal {C} be any category, and let \mathcal {C}^\to denote the arrow category. Then the codomain functor \mathcal {C}^\to \to \mathcal {C} is a fibration if and only if \mathcal {C} admits all pullbacks, and in this case the functors f^*: \mathcal {C}_Y \to \mathcal {C}_X, given f: X \to Y, are given by pullback along f.

The functors f^* are sometimes referred to as pullback, a convention we generally adopt. They are also sometimes called base-change functors.

Let \mathcal {A}: \mathcal {C}^\mathrm {op} \to \mathsf {Cat} be a pseudofunctor. Then there is a category \int _{X \in \mathcal {C}}\mathcal {A}(X) defined as follows:

1. The objects are pairs {\bar {X}\in \mathcal {A}(X) \choose X \in \mathcal {C}}
2. The morphisms {\bar {X} \choose X} \to {\bar {Y} \choose Y} are pairs f: X \to Y, f^\#: \bar {X} \to \mathcal {A}(f)(\bar {Y}) \in \mathcal {A}(X)
3. Composition is given by the "chain rule" (f,f^\#) \circ (g,g^\#) = (fg, \mathcal {A}(g)(f^\#)g^\#)

The category \int _X \mathcal {A}(X) is known as the Grothendieck construction of \mathcal {A}

\int _X \mathcal {A}(X) \to \mathcal {C} is a Grothendieck fibration, and the pseudofunctor it induces is equivalent to \mathcal {A}

Let p: \mathcal {D} \to \mathcal {C} be a fibration. Then there exists a category \mathcal {D}^\mathrm {fop}, called the fiberwise opposite of \mathcal {D}, whose objects are the same as \mathcal {D}, and where a morphism X \to Y is a tuple (f: p(X) \to p(Y), f^\#: f^*Y \to X \in \mathcal {D}_{p(X)}).

Let \mathcal {M} be a monoidal category which acts on two categories \mathcal {C}, \mathcal {D}. Then the category of optics \mathsf {Optic}_\mathcal {M}(\mathcal {C},\mathcal {D}) has

1. Objects pairs {A \in \mathcal {D} \choose X \in \mathcal {C}}
2. The set of morphisms {A \choose X} \to {B \choose Y} given by the coend \int ^{M \in \mathcal {M}} \mathcal {C}(X, M \cdot Y) \times \mathcal {D}(M \cdot B,A)
3. Given two optics with representatives (M,f: X \to M \cdot Y,g : M \cdot B \to A), (N, f': Y \to N \cdot Z, g': N \cdot C \to B), their composite is given by (M \otimes N, (1_M \cdot f')f, g (1_M \cdot g')), where we omit coherence morphisms.

When \mathcal {C} is a monoidal category acting on itself by tensor, we write \mathsf {Optic}_\mathcal {C}(\mathcal {C},\mathcal {C}) =: \mathsf {Optic}(\mathcal {C})

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 [riley-optics] the objects are written (X,A), where X is the forwards part. Hedges' work on open games used the binomial notation \binom {X}{A}, but wrote the forwards part on top.

To make the connection to fibrations more natural, we instead write the forwards part on the bottom, {A \choose X}. 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".

Suppose \mathcal {M}, \mathcal {C}, \mathcal {D} are small categories. Then the coend defining \mathsf {Optic}_\mathcal {M}(\mathcal {C},\mathcal {D})\left ( {A \choose X}, {B \choose Y} \right ) is a small colimit of small sets, hence again small. Since clearly the set of objects \operatorname {\mathbf {ob}} \mathcal {C} \times \operatorname {\mathbf {ob}} \mathcal {D} is small, \mathsf {Optic}_\mathcal {M}(\mathcal {C},\mathcal {D}) is again a small category.

However, if \mathcal {M},\mathcal {D},\mathcal {C} are merely assumed to be locally small, we can not guarantee the same is true of \mathsf {Optic}_\mathcal {M}(\mathcal {C},\mathcal {D}), 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 \mathsf {Lens}(\mathcal {C}) 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.

Given \mathcal {M} acting on \mathcal {C}, \mathcal {D}, the category of co-optics, \mathsf {coOptic}_\mathcal {M}(\mathcal {C},\mathcal {D}) has objects pair {A \in \mathcal {D} \choose X \in \mathcal {C}}, and morphisms given by the coend \int ^M \mathcal {C}(X, M \cdot Y) \times \mathcal {D}(M \cdot A, B)

For any monoidal category \mathcal {C}, \mathsf {Optic}_\mathcal {C}\left ({A \choose X}, {I \choose I}\right ) = \mathcal {C}(X,A). One way to think of an optic {A \choose X} \leftrightarrows {B \choose Y} is as a string diagram X \to A, but which has a hole with space for a morphism B \to Y. One inserts such a morphism by composing the optic with the optic {B \choose Y} \leftrightarrows {I \choose I} representing it. This idea of "open diagrams" has been developed in much more detail by Román, Reference [roman-optics-coend].

Suppose \mathcal {M} is semicartesian---in other words, that I \in \mathcal {M} is terminal. Then there is a functor \mathsf {Optic}_\mathcal {M}(\mathcal {C},\mathcal {D}) \to \mathcal {C}, which takes a {A \choose X} to X, and pair \langle f: X \to M \cdot Y, g \rangle to the composite X \to M \cdot Y \to I \cdot Y \cong Y.

In the case of \mathsf {Optic}(\mathcal {M}), the map \mathsf {Optic}(\mathcal {M})({I \choose I}, {A \choose X}) \to \mathcal {M}(I,X) is a bijection.

If \mathcal {C} is Cartesian monoidal, \mathsf {Optic}(\mathcal {C})\left ({A \choose X},{B \choose Y}\right ) \cong \mathcal {C}(X,Y) \times \mathcal {C}(X \times B,A)

The functor {A \choose X} \mapsto X, \mathsf {Lens}(\mathcal {C}) \to \mathcal {C}, is a fibration. The fiber \mathsf {Lens}(\mathcal {C})_X has the following description:

1. Its objects are the objects of \mathcal {C}.
2. A map A \to B \in \mathsf {Lens}(\mathcal {C})_X is a map X \times B \to A \in \mathcal {C}.
3. The composite of X \times B \to A, X \times C \to B is given by composing the two into X \times X \times C \to A, then using the diagonal.
4. Given f: X \to Y, the pullback functor \mathsf {Lens}(\mathcal {C})_Y \to \mathsf {Lens}(\mathcal {C})_X is given by precomposing by f.

If \mathcal {C} moreover admits pullbacks, there is a fibred functor \mathsf {Lens}(\mathcal {C})^\mathrm {fop} \to \mathcal {C}^\to \to \mathcal {C}, which carries an object {A \choose X} to X \times A\xrightarrow {\pi _X} X, and a morphism f:X \times A \to B to the map X \times A \xrightarrow {\langle \pi _X,F \rangle } X \times B over X. This is fully faithful.

The squares

Let \mathcal {C} be a symmetric monoidal category. A premarkov structure on \mathcal {C} is a wide symmetric monoidal subcategory \mathcal {C}' \subseteq \mathcal {C}---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 \mathcal {C}' is Cartesian.

Let \mathcal {C} be a monoidal category.

1. Given a premarkov structure \mathcal {C}' \subseteq \mathcal {C}, there is a unique Markov structure on \mathcal {C} so that each morphism in \mathcal {C}' is deterministic.
2. Given a Markov structure, \mathcal {C}_\mathrm {det} \subseteq \mathcal {C} is a premarkov structure.
3. A premarkov structure has the form \mathcal {C}_\mathrm {det} for some Markov structure if and only if it is maximal.

A Markov category is representable if the inclusion \mathcal {C}_\mathrm {det} \hookrightarrow \mathcal {C} admits a right adjoint. In this case we denote the right adjoint P and call the object PX for X \in \mathcal {C} a distribution object for X. Observe that \mathcal {C}(X,Y) = \mathcal {C}_\mathrm {det}(X,PY) (by definition,) and hence \mathcal {C} = Kl(P) where we denote the induced monad on \mathcal {C}_\mathrm {det} P by an abuse of notation.

Here are some important Markov categories:

1. \mathsf {Stoch} (Reference [fritz-synthetic-markov-cats], 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.
2. \mathsf {BorelStoch} \subseteq \mathsf {Stoch} (Reference [fritz-synthetic-markov-cats], section 4) is the full subcategory of \mathsf {Stoch} spanned by the standard Borel spaces, that is by those measurable spaces arising as the Borel \sigma -algebra on separable, complete metric space.
3. \mathsf {FinStoch} is the subcategory of \mathsf {Stoch} spanned by finite sets in the powerset \sigma -algebra. A morphism X \to Y in \mathsf {FinStoch} is equivalently a matrix f_{xy} : x \in X, y \in Y with entries in \mathbb {R}_{\geq 0} and with \sum _y f_{xy} = 1 for each x (this is what is called a stochastic matrix).
4. Let \Delta : \mathsf {Set} \to \mathsf {Set} be the monad which assigns to X \in \mathsf {Set} the set \Delta (X) of countably-supported probability measures. Then the Kleisli category Kl(\Delta ) is a Markov category, sometimes called the category of discrete probability.
5. \mathsf {TychStoch} (Reference [markov-supports], example A.1.4) is the category of Tychonoff topological spaces and kernels which are valued in Radon probability measures, and where the measure f(- \mid x) varies continuous in x \in X with respect to the weak topology---in other words, given any continuous function u \in C(Y), the resulting function on X given by E_{y \sim f(- \mid x)}u(y) is continuous. (A space X is Tychonoff if it is Hausdorff and, given K \subset X closed and x_0 not in K, there exists continuous f: X \to [0,1] with f(x_0)=0, f(k) =1 for k \in K. Every locally compact Hausdorff space is Tychonoff).

If \mathcal {C} is a Markov category and I is any ordinary category, there is a Markov category \mathsf {Fun}(I,\mathcal {C}) whose objects are functors I \to \mathcal {C}_\mathrm {det}, and whose morphisms are natural transformations between these considered as functors into \mathcal {C} (i.e natural transformations with stochastic components). The monoidal and Markov structure is defined simply component-wise.

In particular, taking I = \to = \{0 \to 1\} the walking arrow, we obtain a Markov category of deterministic arrows \mathsf {Fun}(\to , \mathcal {C}). We will denote this category simply \mathcal {C}^\to . Again, the objects of this category are the deterministic morphisms of \mathcal {C}, while the morphisms are the commutative squares with not-necessarily-deterministic sides

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 \mathcal {C}.

Given a map p: P \to X \otimes Y into a tensor product, the composites with the projections P \to X, P \to Y are called the marginals. Given two maps f: P \to X, g: P \to Y, we refer to any map P \to X \otimes Y with those marginals as a pairing of the two. Note that there is always a canonical pairing given by (f \otimes g) \mathrm {copy}_P. We write \langle f,g \rangle : P \to X \otimes Y 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 p = \langle \pi _Xp, \pi _Yp \rangle , we say X \bot Y \mid P, and say the two coordinates are independent given P (or just independent).

Let \mathcal {C} be a Markov category. The following are equivalent:

1. In \mathcal {C}, if f: P \to X \otimes Y has deterministic marginal P \to X, then X \bot Y \mid P. In other words, anything is independent of a deterministic variable.
2. Given a deterministic morphism P \to X and any morphism P \to Y, there is a unique P \to X \otimes Y with those marginals.
3. \mathcal {C} is positive.

Let \mathcal {C} be a Markov category. We say \mathcal {C} is pullback-positive if \mathcal {C}_\mathrm {det} admits pullbacks and, given a diagram of this form where P \to X is deterministic, there is a unique map P \to X \times _Z Y making the squares commute.

The pullbacks appearing in Definition [efr-T3HM] are of course pullbacks in \mathcal {C}_\mathrm {det}, not \mathcal {C}, analogously to how X \otimes Y is a product in \mathcal {C}_\mathrm {det}, not \mathcal {C}. We will still use the notation X \times _Z Y for these pullbacks, which should not lead to any confusion. We may occasionally write X \times Y instead of X \otimes Y, if we are carrying out a construction which primarily involves \mathcal {C}_\mathrm {det}. Since essentially no Markov categories have Cartesian products (except when the tensor product is Cartesian), this should also not lead to any ambiguity.

1. Suppose any category \mathcal {C}' admits products and intersections---that is, pullbacks U \times _X V whenever U,V are subobjects of X. Then it admits all finite limits.
2. Suppose \mathcal {C}_\mathrm {det} admits and \mathcal {C}_\mathrm {det} \to \mathcal {C} preserves pullbacks along monomorphisms. Suppose further \mathcal {C} is positive. Then it is pullback-positive.

If \mathcal {C} is representable and positive, \mathcal {C}_\mathrm {det} has limits, and the monad P preserves intersections, then \mathcal {C} is pullback-positive.

Let p: X \to Y, q: A \to Y be two morphisms in a Markov category. We say q is absolutely continuous with respect to p and write q << p if, whenever two maps Y \to Z are p-almost surely equal, they are also q-almost surely equal.

Let p: X \to Y be a morphism in a Markov category. The support of p, if it exists, is an object which represents the functor \mathcal {C}(-,Y)_{<<p} of morphisms into Y which are absolutely continuous with respect to p.

1. The support of p: X \to A is equipped with a canonical deterministic monomorphism S_p \hookrightarrow A, so that a map into A is absolutely continuous with respect to p if and only if it factors over the support.
2. Two maps A \otimes W \to B are p-almost surely equal if and only if they are strictly equal on the support.

A (strict) double category is a category internal to the category \mathsf {Cat} of categories. Concretely, it consists of:

1. A set of objects \operatorname {\mathbf {ob}} \mathbb {C}
2. A collection of vertical morphisms forming a category \mathbb {C}_v with \operatorname {\mathbf {ob}} \mathbb {C}_v = \operatorname {\mathbf {ob}} \mathbb {C}
3. A collection of horizontal morphisms forming a category \mathbb {C}_h with \operatorname {\mathbf {ob}} \mathbb {C}_h = \operatorname {\mathbf {ob}} \mathbb {C}
4. An a collection of squares. Each square has a left and right boundary given by vertical morphisms l,r, and top and bottom boundary given by horizontal morphisms t,b so that \operatorname {\mathrm {dom}} t = \operatorname {\mathrm {dom}} l, \operatorname {\mathrm {cod}} t = \operatorname {\mathrm {dom}} r 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 a pseudo double category \mathsf {\mathbb Cat} where the objects are categories, the vertical maps are functors, the horizontal maps are profunctors (functors \mathcal {C} \times \mathcal {D}^\mathrm {op} \to \mathsf {Set}, sometimes called bimodules), and the squares are natural transformations.

For any category with pullbacks \mathcal {C}, there is a pseudo double category \mathsf {\mathbb Span}(\mathcal {C}) with \mathcal {C} as the vertical category, spans as the horizontal morphisms, and commutative diagrams as the squares.

There is a double category \mathsf {\mathbb Rel} of sets, functions, and relations.

For any Markov category \mathcal {C}, there is a double category with \mathbb {C}_v = \mathcal {C}_\mathrm {det} and \mathbb {C}_h = \mathcal {C}.

For any category at all, there is a double category \operatorname {Sq}(\mathcal {C}) with \operatorname {Sq}(\mathcal {C})_v = \operatorname {Sq}(\mathcal {C})_h = \mathcal {C} and commutative squares as the pullback squares.

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.

A theory of dynamical systems is an indexed category \mathcal {A}: \mathcal {C}^\mathrm {op} \to \mathsf {Cat} equipped with a section of its Grothendieck construction T: \mathcal {C} \to \int \mathcal {A}.

It is called monoidal if \mathcal {C} is a monoidal category, and \mathcal {A} is a lax monoidal functor. It is further called symmetric monoidal if \mathcal {C} is a symmetric monoidal category and \mathcal {A} is a symmetric lax monoidal functor. Note that this is equivalent to requiring \int \mathcal {A} \to \mathcal {C} to be a (symmetric) monoidal fibration in the sense of Reference [shulman-monfibs] (see also Reference [moeller-vasilakopoulou]).

Let \mathcal {A}: \mathcal {C}^\mathrm {op} \to \mathsf {Cat} be an indexed category. A chart is simply a morphism in the Grothendieck construction \int \mathcal {A}. A lens is a morphism in the fiberwise opposite \int \mathcal {A}(-)^\mathrm {op}.

Given an indexed category \mathcal {A}: \mathcal {C}^\mathrm {op} \to \mathsf {Cat}, the double category \mathsf {\mathbb Arena} of arenas has

1. Objects the objects of \int \mathcal {A}---note that these are the same as the objects of \int \mathcal {A}(-)^\mathrm {op}
2. Vertical morphisms the lenses
3. Horizontal morphisms the charts
4. The double category is thin. Given a square of this form

   

   we can first project it to a square in \mathcal {C}. If this commutes, we can pull the lenses and charts back to a square in \mathcal {A}(A_1). The above square of lenses and charts will be said to commute if both of these squares commute.

Let A be an arena. A dynamical system with interface A is an object S \in \mathcal {C} and a lens TS \leftrightarrows A. A morphism of systems is a map S \to S' so that this square commutes in \mathsf {\mathbb Arena}:

The category of systems with interface A is denoted \mathsf {Sys}(A)

Given a lens A \leftrightarrows A', postcomposition gives a functor \mathsf {Sys}(A) \to \mathsf {Sys}(A'). Given a chart A \rightrightarrows A', there is a profunctor \mathsf {Sys}(A) \nrightarrow \mathsf {Sys}(A'), where the set over TS \leftrightarrows A and TS' \leftrightarrows A' is the set of maps S \to S' so that this square commutes:

This defines a double functor \mathsf {Sys}: \mathsf {\mathbb Arena} \to \mathsf {\mathbb Cat}.

Let F: \mathsf {Set} \to \mathsf {Set} be a functor. Then there is a dynamical systems theory where \mathcal {C} = \mathsf {Set}, \mathcal {A} is constant at \mathsf {Set}, and T(X) = (X,F(X)). It is easy to see that the category of closed dynamical systems in this theory is equivalently the category of F-coalgebras.

There is a theory of dynamical systems with category of spaces \mathcal {C} = \mathsf {Set}, category of bundles \mathcal {A}(X) = \mathsf {Set}_{/X} (with pullbacks for reindexing), and tangent bundle TX = X \times X \xrightarrow {\pi _0} X. We call this the theory of discrete dynamical systems. (In the sense that they are both discrete-time and discrete-space).

In the theory of discrete dynamical systems:

1. The category of lenses \int \mathcal {A}(-)^\mathrm {op} is equivalently the category of polynomial functors and natural transformations between them.
2. Using this equivalence, the category of dynamical systems with interface p is exactly the category of p-coalgebras and homomorphisms

There is a systems theory where \mathcal {C} = \mathsf {Set}, \mathcal {A}(X) = Kl(\Delta )^X (with the evident reindexing maps) and T(X) = X \times X.

In this theory, a closed dynamical system is a set equipped with a map X \to \Delta (X). An open dynamical system has stochastic update function, but output which depends deterministically on the current system state.

There is a dynamical systems theory where \mathcal {C} = \mathsf {SmMfd} is the category of smooth manifolds, \mathcal {A}_X is the category of fiber bundles over X, with f^*: \mathcal {A}_Y \to \mathcal {A}_X given by pullback along f: X \to Y (note that pullbacks of bundles always exist), and with TX being the tangent bundle of X 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.

Consider the theory of discrete-time, discrete dynamical systems from Example [efr-002F]. Let \xi : TS \leftrightarrows {I \choose O} be a system. Consider the system c: T\mathbb {N} \leftrightarrows {* \choose \mathbb {N}} given by n \mapsto n in the forwards direction, and (n,*) \mapsto n+1 in the backwards direction. (To be clear, the object denoted {* \choose \mathbb {N}} is the map \mathbb {N} \to \mathbb {N}---it is a singleton in each fiber). Then a morphism of systems c \to \xi consists of the following data:

1. A function x: \mathbb {N} \to S.
2. Another function o: \mathbb {N} \to O.
3. A third function i: \mathbb {N} \to I so that i(n) \in I_{o(n)}.
4. So that o(n) = \xi (x(n)) and \xi ^\#(x(n),i(n)) = x(n+1)

That is, it is a choice of a sequence x_n of points in the state-space, and a sequence of inputs i_n compatible with the outputs, so that this sequence obeys the dynamics. In other words, it is a trajectory of the system.

There exists a (strong) symmetric monoidal functor \mathsf {Optic}(Kl(\Delta )) \to \mathsf {SLens}(Kl(\Delta )^\to ) which

1. Is fully faithful.
2. Preserves the good coproducts.
3. So that the image has all coproducts, and every object of the image is a coproduct of objects in \mathsf {Optic}(Kl(\Delta ))
4. And so that moreover the monoidal structure on the image is distributive

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 \times _Y X, a map P \to A \times _Y X where the X-coordinate is deterministic should be uniquely determined by a choice of (deterministic) map P \to X and (stochastic) P \to A such that the square commutes---as we claimed above (and will see below), this holds for the Markov category of discrete probability Kl(\Delta ). The analogous statement for products---that a map P \to A \otimes X with deterministic X-component is uniquely determined by the projections (or marginals) P \to X, P \to A---is a consequence of positivity (see Proposition [efr-OYB6]), and hence holds in most Markov categories of interest.

Let \mathcal {C} be a Markov category, and let p: \mathcal {D} \to \mathcal {C} be a functor into it. Then we call p a Markov prefibration if the following two conditions hold:

1. The pullback \mathcal {D} \times _\mathcal {C} \mathcal {C}_\mathrm {det} \to \mathcal {C}_\mathrm {det} is a (Grothendieck) fibration
2. Given maps f: A \to C, g:B \to C in \mathcal {D}, such that p(f),p(g) are deterministic and f,g are Cartesian for the above fibration, p induces a bijection between maps h: A \to B \in \mathcal {C} such that gh = f, and maps h': p(A) \to p(B) so that p(g)h' = p(f). Note that when restricted to those maps where p(h) is deterministic, this being a bijection is the defining property of g being Cartesian (for any f, not necessarily a Cartesian one).

Given a Markov prefibration \mathcal {D}, we write \mathcal {D}|_\mathrm {det} for \mathcal {D} \times _\mathcal {C} \mathcal {C}_\mathrm {det}. We will refer to this as the deterministic part of \mathcal {D}---note that this does have the potential for confusion, as when \mathcal {D} is itself a Markov category, this is not necessarily the same as the deterministic subcategory of \mathcal {D}. When f \in \mathcal {D} lies inside \mathcal {D}|_\mathrm {det}, and is Cartesian for that fibration, we will simply refer to it as a Cartesian map in \mathcal {D} (there are no other types of Cartesian maps, so this should not lead to confusion). A morphism of Markov prefibrations is a functor \mathcal {D} \to \mathcal {D}' over \mathcal {C} which preserves Cartesian maps. The category of Markov prefibrations over \mathcal {C} thus defined is denoted \mathsf {MarkPreFib}(\mathcal {C}). Taking the deterministic part defines a functor (-)|_\mathrm {det}: \mathsf {MarkPreFib}(\mathcal {C}) \to \mathsf {Fib}(\mathcal {C}_\mathrm {det}).

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 (-)|_\mathrm {det}: \mathsf {MarkPreFib}(\mathcal {C}_\mathrm {det}) \to \mathsf {Fib}(\mathcal {C}), 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 \mathcal {C}_\mathrm {det} \to \mathcal {C} 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 (-)|_\mathrm {det} is automatically strict---it simply consists in restriction to a subset of the morphisms in \mathcal {D} (which is automatically closed under composition), and thus clearly preserves composition strictly.

In a Markov prefibration p: \mathcal {D} \to \mathcal {C} (as previously noted), a morphism f: \bar {X} \to \bar {Y} in \mathcal {D} is called Cartesian if p(f) is deterministic and f is Cartesian in the fibration \mathcal {D}|_\mathrm {det} \to \mathcal {C}_\mathrm {det}. It is called vertical if p(f) is an identity. It is called a stochastic-Cartesian if there exists a Cartesian map r: \bar {Y} \to \bar {X} so that rf = 1_{\bar {X}} (recall that in this case f is uniquely determined by r and p(f)). Note that if f is stochastic-Cartesian and p(f) is deterministic, then f is Cartesian.

Let \mathcal {C} be a Markov category which is Cartesian (that is, one where all morphisms are deterministic). Then a Markov prefibration over \mathcal {C} is simply a Grothendieck fibration.

Let \mathcal {C} be a Markov category. Then \mathcal {C}^\to \to \mathcal {C} is a Markov prefibration with Cartesian maps given by pullback squares, if and only if \mathcal {C} is pullback-positive.

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 [rischel-fritz-infinite-products], 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 \mathcal {C}, such as positivity, all isomorphisms are deterministic. This implies that all Markov prefibrations over \mathcal {C} 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 [efr-0044], 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).

Let p: \mathcal {D} \to \mathcal {C} be a Markov prefibration, let \mathcal {C}' \subseteq \mathcal {C}, \mathcal {D}' \subseteq \mathcal {D} be full subcategories so that p(\mathcal {D}') is contained in \mathcal {C}', and suppose \mathcal {C}' is a monoidal subcategory (which is then automatically a sub-Markov category). Suppose finally \mathcal {D}' is stable under pullback along deterministic morphisms in \mathcal {C}'. Then \mathcal {D}' \to \mathcal {C}' is again a Markov prefibration.

Recall that the deterministic maps in \mathsf {Stoch} are those kernels valued in \{0,1\}-valued measures. (And that these are not the same thing as the measurable maps). Given a pair f:X \to Z, g: Y \to Z of such maps, we claim that the subset of X \times Y given by those points where f(x) = g(y) as measures on Z, equipped with the subset \sigma -algebra and its inclusions into X and Y, is a pullback in \mathsf {Stoch}_\mathrm {det}.

To see this, observe that \mathsf {Stoch}_\mathrm {det} is the Kleisli category for the submonad G_{0,1} of the Giry monad which takes a space to the subspace of 0,1-valued probability measures on it. Note that by the general theory of Markov categories this has products given by the products in \mathsf {Meas}. It hence suffices to prove that the pullback of f \times g: X \times Y \to Z \times Z along the diagonal Z \to Z \times Z exists---if it does, it has the universal property of the product X \times _Y Z.

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 h: P \to X \times Y \in \mathsf {Stoch}_\mathrm {det} factors over X \times _Y Z if and only if the composite to Z \times Z lifts over the diagonal.

Now this lift over Z exists if and only if the probability of the diagonal \operatorname {im}(Z) \subseteq Z \times Z under the composite is 1. By definition, it is \int f(\operatorname {im}(Z) \mid x) g(\operatorname {im}(Z) \mid y) h(dx,y \mid p) Since the function being integrated is an indicator, this is simply the measure of the set \{(x,y) : \mathbb {P}(\operatorname {im}(Z) \mid x,y) = 1\}. If the probability of the diagonal is one, clearly the two marginals agree. Conversely, since \mathsf {Stoch}_\mathrm {det} 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 X \times _Z Y \subseteq X \times Y described at the start. But for this to have measure 1 for all p \in P is equivalent to the desired lifting property. This finishes the argument.

Moreover, this argument relied only on the 0,1-valuedness of the maps f,g, not h: P \to X \times Y or its marginals. Hence this also proves that the pullback along the diagonal is preserved by the inclusion \mathsf {Stoch}_\mathrm {det} \to \mathsf {Stoch}. Since \mathsf {Stoch} is known to be positive, this implies \mathsf {Stoch}^\to \to \mathsf {Stoch} is a Markov prefibration.

The codomain functor \mathsf {BorelStoch}^\to \to \mathsf {BorelStoch} is a Markov prefibration.

\mathsf {FinStoch}^\to \to \mathsf {FinStoch} is a Markov prefibration.

The functor \mathsf {Optic}(\mathsf {BorelStoch}) \to \mathsf {BorelStoch} is not a Markov prefibration, although its pullback over \mathsf {BorelStoch}_\mathrm {det} is a Grothendieck fibration.

To see this, first consider the deterministic pullback. An optic \binom {A}{X} \to \binom {B}{Y} with deterministic base can be identified with a map X \otimes B \to A (and the base deterministic map X \to Y). To see this, first observe that the subset of \mathsf {BorelStoch}(X, Y \otimes M) with the marginal X \to Y deterministic is in bijection with \mathsf {BorelStoch}_\mathrm {det}(X,Y) \times \mathsf {BorelStoch}(X,M), since \mathsf {BorelStoch} is positive. Hence we can calculate \int ^M \mathsf {BorelStoch}_\mathrm {det}(X,Y) \times \mathsf {BorelStoch}(X,M) \times \mathsf {BorelStoch}(M \otimes B, A) \cong \mathsf {BorelStoch}_\mathrm {det}(X,Y) \times \mathsf {BorelStoch}(X \times B, A), using the ninja yoneda lemma as in Proposition [efr-M19V]. Hence this part is a fibration with the fiber over X being the coKleisli category of the X \times - monad, and the pullback functors given by reindexing these parametrized maps. The Cartesian lift of a map X \to Y at \binom {B}{Y} is given by the optic \binom {B}{X} \to \binom {B}{Y} with unit residual and identity backwards component.

Now, let g: I \to \mathbb {R} denote the standard Gaussian distribution, let f: \mathbb {R} \otimes \mathbb {R} \to \mathbb {R} be the function given by f(x,y) = 0 if x=y and y otherwise, and consider the two optics \binom {\mathbb {R}}{*} \to \binom {\mathbb {R}}{\mathbb {R}} given by (I, g: I \to \mathbb {R}, 1_\mathbb {R}: \mathbb {R} \to \mathbb {R}), (\mathbb {R}, \mathrm {copy}_\mathbb {R} g : I \to \mathbb {R} \otimes \mathbb {R}, f: \mathbb {R} \otimes \mathbb {R} \to \mathbb {R}) (where we recall that the first argument is the residual). They cannot be equal, as postcomposition with the optic \binom {\mathbb {R}}{\mathbb {R}} \to \binom {*}{*} given by the identity \mathbb {R} \to \mathbb {R} yields, for the former, the standard Gaussian g: I \to \mathbb {R}, and for the latter, the constant zero map. But postcomposition with the projection \binom {\mathbb {R}}{\mathbb {R}} \to \binom {\mathbb {R}}{*} does give the same optic (the identity), because, for every fixed y \in \mathbb {R}, when x is normally distributed, f(x,y) = y with probability one. Hence the unique lifting of Cartesian maps over Cartesian maps cannot hold.

Let \mathcal {C} be a positive Markov category with supports. Then \mathsf {Optic}(\mathcal {C}) \to \mathcal {C} is a Markov prefibration.

Let p: \mathcal {D} \to \mathcal {C} be a functor. An indexed copresheaf on \mathcal {D} is a tuple (X \in \mathcal {C}, F: \mathcal {D} \to \mathsf {Set}, \alpha : F(-) \to \mathcal {C}(X,p(-))) consisting of a copresheaf, an object of \mathcal {C}, and a natural transformation \alpha as indicated. We say the indexed copresheaf is over X, and we will abuse the terminology by referring to F itself as an indexed copresheaf, leaving the transformation \alpha implicit (for example, "let F be an indexed copresheaf over X").

We denote the subset \alpha ^{-1}(\{f\}) \subseteq F(\bar {A}), for f: X \to p(\bar {A}) by F(\bar {A})_f.

A map of indexed copresheaves (X,F,\alpha ) \to (Y,G,\beta ) is a natural transformation F \to G and a map Y \to X \in \mathcal {C} so that the obvious square of natural transformations commutes. We denote the category of indexed copresheaves by \mathsf {IcoPSh}(\mathcal {C} / \mathcal {D}). Note that there is an obvious forgetful functor \mathsf {IcoPSh}(\mathcal {D} / \mathcal {C})^\mathrm {op} \to \mathcal {C}

Observe that for each object A \in \mathcal {D}, there is a corepresentable copresheaf (p(A), \mathcal {D}(A,-), p). Maps between these obey the Yoneda lemma, in the sense that they are in bijection with maps between the underlying objects in \mathcal {D}. This defines a fully faithful functor \mathcal {D} \to \mathsf {IcoPSh}(\mathcal {D} / \mathcal {C})^\mathrm {op} over \mathcal {C}.

Let p: \mathcal {D} \to \mathcal {C} be any functor and let \mathcal {C}_0 \to \mathcal {C} be an identity-on-objects functor. Write \mathcal {D}_0 = \mathcal {D} \times _\mathcal {C} \mathcal {C}_0 for the pullback. If F: \mathcal {D} \to \mathsf {Set} is a copresheaf indexed over X \in \mathcal {C}, the pullback \bar {A} \mapsto F(\bar {A}) \times _{\mathcal {C}(X,p\bar {A})} \mathcal {C}_0(X,p\bar {A}) is a copresheaf on \mathcal {D}_0 indexed over X again in a unique way. This defines a functor \mathsf {IcoPSh}(\mathcal {D} / \mathcal {C}) \to \mathsf {IcoPSh}(\mathcal {D}_0 / \mathcal {C}_0). Moreover, this functor preserves the corepresentable copresheaves (since \mathcal {C}_0 \to \mathcal {C} is identity on objects, so is \mathcal {D}_0 \to \mathcal {D}, so this statement makes sense).

Let \mathcal {D}_0 \to \mathcal {C}_0 be a functor and let \mathcal {C}_0 \to \mathcal {C} be identity-on-objects and faithful. Then the pullback of the composite \mathcal {D}_0 \to \mathcal {C} and the inclusion \mathcal {C}_0 \to \mathcal {C} is identical to \mathcal {D}_0. Therefore, Proposition [efr-FOJT] gives a functor \mathsf {IcoPSh}(\mathcal {D}_0 / \mathcal {C}) \to \mathsf {IcoPSh}(\mathcal {D}_0 / \mathcal {C}_0).

Let \mathcal {C} be a Markov category, and let \mathcal {D}_0 \to \mathcal {C}_\mathrm {det} be a fibration. Then a stochastic module consists of

1. An indexed copresheaf F = (A,F,\rho ) \in \mathsf {IcoPSh}(\mathcal {D}_0 / \mathcal {C})
2. For each \bar {X} \in \mathcal {D}_0, Cartesian morphism \bar {a}: \bar {X}_M \to \bar {X} lying over a: M \to X, and stochastic section \alpha : X \to M, a function \alpha _*: F(\bar {X}) \to F(\bar {X}_M), which acts on the underlying morphisms in \mathcal {C} as composition with \alpha
3. Satisfying, whenever given a commutative square of Cartesian morphisms:

   

   and stochastic sections \alpha ,\beta lying over a = p(\bar {a}), b = p(\bar {b}), so that we have a digram in \mathcal {C}:

   

   Where the maps except \alpha ,\beta are deterministic, \alpha and \beta are sections of a and b, and both the square of deterministic maps and the square involving \alpha ,\beta commute, the condition that the square

   

   commutes.
4. And satisfying furthermore the equation, for every two stochastic sections \alpha :X \to M, \beta : M \to N, the equation (\beta \alpha )_* = \beta _*\alpha _* (note that this makes sense because pullbacks compose).

A morphism of stochastic modules is an indexed natural transformation which preserves the operations \alpha _*. The category of stochastic modules is denoted \mathsf {SMod}(\mathcal {D}_0 / \mathcal {C}). There is an apparent forgetful functor \mathsf {SMod}(\mathcal {D}_0 / \mathcal {C}) \to \mathsf {IcoPSh}(\mathcal {D}_0 / \mathcal {C}).

Note that \alpha _* of course depends on f, not just \alpha .

If f: M \to X is a deterministic map, \alpha :X \to M is a stochastic section, and \bar {X}_M, \bar {X}_M' \to \bar {X} are two Cartesian lifts of f to \bar {X} \in \mathcal {D}_X, then applying the commutativity axiom for stochastic modules implies that the triangle

Let \mathcal {D} \to \mathcal {C} be a Markov prefibration, and let \mathcal {D}_0 = \mathcal {D} \times _\mathcal {C} \mathcal {C}_\mathrm {det}. Then the corepresentable copresheaf \mathcal {D}(\bar {A},-), restricted to \mathcal {D}_0, (but not pulled back---that is, we remember the whole set \mathcal {D}(\bar {A},\bar {X}), even the part over stochastic f, but only the composition with maps in \mathcal {D}_0) is a stochastic module in a canonical way, with \alpha _*: F(\bar {X}) \to F(f^*\bar {X}) given by composition with the unique induced lift of \alpha . Moreover, any morphism of Markov prefibrations \phi : \mathcal {D} \to \mathcal {D}' induces a homomorphism of stochastic modules \mathcal {D}(\bar {A},-) \to \mathcal {D}'(\phi (\bar {A}),-)

1. Let \bar {X} \in \mathcal {D}_0 be an object. Then there is a free stochastic module T\mathcal {D}_0(\bar {X},-) on its representable copresheaf, in the sense that if F is another stochastic module, homomorphisms T\mathcal {D}_0(\bar {X},-) \to F are in bijection with indexed natural transformations \mathcal {D}_0(\bar {X},-) \to F
2. The free stochastic module on a corepresentable presheaf is given as follows: an element of T\mathcal {D}_0(\bar {X},-)(\bar {Y}) consists of a diagram in \mathcal {C} of the form

   

   (where X = p(\bar {X}), Y = p(\bar {Y}),) where fs = 1_X and f,g are deterministic, plus a map f^*\bar {X} \to \bar {Y} lying over g. This is up to the equivalence relation which, given some other such tuple, identifies them whenever there exists deterministic h: N \to M as in this diagram:

   

   so that the two deterministic triangles commute, hs' = s, and so that the unique Cartesian map f'^*\bar {X} \to f^*\bar {X} over h forms a commutative triangle with the two maps to \bar {Y}. (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 \bar {Y} \to \bar {Z} acts on this to make it a copresheaf. It is indexed by taking a tuple as above to the composite X \to M \to Y. Given N \to Y with a stochastic section s': Y \to N,, and an element of T\mathcal {D}_0(\bar {X},-)(\bar {Y}), the induced element in T\mathcal {D}_0(\bar {X},-)(s'^*\bar {Y}) is given by forming the pullback M \times _Y N, taking the pullback of the map over g to one lying over the projection M \times _Y N \to N, and composing the section s with the induced lift M \to M \times _Y N

   

3. The underlying copresheaf of this respects Cartesian maps in \mathcal {D}_0 \to \mathcal {C}_\mathrm {det}, in the sense that given a Cartesian map \bar {A} \to \bar {B} and an element \phi \in T\mathcal {D}_0(\bar {X},-)(\bar {B}) lying over a deterministic map X \to B, the natural map from lifts \psi \in T\mathcal {D}_0(\bar {X},-)(\bar {A}) to lifts X \to A is a bijection.

Proof

• March 27, 2025
• Eigil Fjeldgren Rischel

First, we have to verify the equations of a stochastic module for T\mathcal {D}_0(\overline {X},-). For the first case, suppose we are given a square

with all but the upwards maps stochastic. Now let \bar {Z} be some object over Z and suppose we are given an element of T\mathcal {D}_0(\bar {X}, \bar {Z}_Y), represented by a span X \leftarrow M \to Y, a section X \to M and a map \phi : \bar {X}_M \to \bar {Z}_M over M. Consider then the below diagram:

By definition, the first of the two possible elements of T\mathcal {D}_0(\bar {Z}_W) are given by either forming the pullback M \times _Y N, taking the lift of \alpha to M \to M \times _Y N and composing to get a section X \to M \times _Y N, and pulling back \phi along the projection to get a map \bar {X}_{M \times _Y N} \to \bar {Z}_{M \times _Y N}, then taking the span X \leftarrow M \times _Y N \to W. The second is given by first composing with the map Y \to Z, then applying the above procedure with the pullback M \times _Z W. The induced map M \times _Y N \to M \times _Z W exhibits the equality of these two under the equivalence relation defining T\mathcal {D}_0(\bar {X},-) (commutativity of the bottom-right square implies that triangle of stochastic sections commutes.)

Given some other stochastic module F over A with a map of indexed copresheaves \mathcal {D}_0(\bar {X},-) \to F, over A \to X, there is at most one extension to a map of stochastic presheaves T\mathcal {D}_0(\bar {X},-) \to F over A \to X---given an element with representative (s:X \to M, X \leftarrow M \to Y, \bar {X}_M \to \bar {Y}_M), it must go to the identity element of F(\bar {X}), acted on by the stochastic section s to produce an element of F(\bar {X}_M), followed by F applied to the map \bar {X}_M \to \bar {Y} over M \to Y.

But it is not hard to see that the equivalence relation imposed by T\mathcal {D}_0(\bar {X},-) is implied by the equations of a stochastic module, and so this map is well-defined, establishing the property.

Secondly, let \bar {A} \to \bar {B} be a Cartesian map over A \to B, and take a commutative triangle

of deterministic maps. Finally take an element of T\mathcal {D}_0(\bar {X},-)(\bar {B}) over the given map X \to B. We must show it has a unique lift to A over the given map X \to A. Let us take a representative given by a diagram:

First, note that the two maps M \to X \to B and M \to B do not necessarily agree. However, we can remedy this by replacing M by their equalizer---note that as we argued above, this gives an equivalent element of the stochastic module. (By writing their equalizer in \mathcal {C}_\mathrm {det} as the split pullback M \times _{B \times B} B, we can see that the section factors over this, even if \mathcal {C} does not have all equalizers in general). Hence we can assume the triangle formed by adding the dashed arrow commutes.

Since M \to B now factors over X, the lift X \to A gives a lift M \to A. Now by the Cartesian property of \bar {A} \to \bar {B}, there is a unique lift of p^*\bar {X} \to \bar {B} to this map (here we just use the fact that \mathcal {D}_0 is a fibration). This gives the desired lift.

Finally, given two distinct lifts (again, we can assume their maps M \to A factor over X,) clearly any map N \to M witnessing an identity between their composites \bar {X} \to \bar {Y} would likewise exhibit an identity between their lifts (since pullbacks compose). This proves uniqueness, and finishes the proof.

Let \mathcal {C} be a pullback-positive Markov category and let \mathcal {D}_0 \to \mathcal {C}_\mathrm {det} be a fibration. Consider the full subcategory of stochastic modules spanned by the free modules on the corepresentables. Denote the opposite of this category \bar {\mathcal {D}_0}. Clearly there is a commutative diagram

1. \bar {\mathcal {D}_0} \to \mathcal {C} is a Markov prefibration.
2. There is a bijection \bar {\mathcal {D}_0}(\bar {A},-) \cong T(\mathcal {D}_0(\bar {A},-)). 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.
3. \bar {\mathcal {D}_0} \to \mathcal {C} is initial among Markov prefibrations receiving a map from \mathcal {D}_0. In other words, this construction gives a left adjoint to the pullback functor \mathsf {MarkPreFib}(\mathcal {C}) \to \mathsf {Fib}(\mathcal {C}_\mathrm {det})

We will sometimes refer to the morphisms of \overline {\mathcal {D}_0} as precharts. Taking the fibration \mathcal {C}^\to |_\mathrm {det} \to \mathcal {C}_\mathrm {det} as an example, it is not too hard to see that the precharts between X \otimes A \to X and B \otimes Y \to Y are representatives of co-optics {A \choose X} \rightrightarrows {B \choose Y}. (To see this, note that any prechart is equivalent to one where the apex of the span has the form M \otimes Y and the right leg is the projection to Y. Then the rest of the data is a map X \to M \otimes Y and a map M \otimes B \to A, since the X-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 \mathsf {Optic}_{\mathcal {C}_\mathrm {det}}(\mathcal {C}_\mathrm {det}, \mathcal {C})). The precharts in \mathcal {D}_0^\mathrm {fop} will be called prelenses. We will speak of the tuple (M, p:M \to X, p':M \to Y, s: X \to M, \phi : p^*\bar {X} \to p^*\bar {Y}) 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 (M,s,\phi ), or even just (M,\phi ). It will be clear from context which part of the structure is being specified.

Let \mathcal {D}_0 be a fibration. Then the underlying fibration of the free Markov prefibration, \overline {\mathcal {D}_0}|_\mathrm {det}, has fiber over X \in \mathcal {C} given by

1. Objects are simply objects of \mathcal {D}_{0,X}
2. A morphism \bar {X} \to \bar {X}' consists of a deterministic f: M \to X, a stochastic section s: X \to M, and a map \phi : f^*\bar {X} \to f^*\bar {X}', up to the equivalence relation generated by, whenever g: N \to M is deterministic and s': X \to N is a factorization of s, identifying (M,f,s,\phi ) with (N,fg, s', g^*(\phi )).
3. Given two such morphisms (M,f,s,\phi ), (N,f',s',\psi ), their composite is represented by M \times _X N \to X equipped with the section formed as the composite of X \to M and the lift of X \to N to the pullback, and the composite \pi _M^*(\phi )\pi _N^*(\psi ) \in \mathcal {D}_{0,M\times _X N}
4. Given deterministic f: X \to Y, the pullback is given on such a map by taking the pullback M \times _Y X \to X, the induced section, and the pullback of the map \phi along the projection M \times _Y X \to M

The free Markov prefibration monad \overline {(-)}|_\mathrm {det} commutes with fiberwise opposites. In particular, algebra structures on \mathcal {D}_0 are in bijection with algebra structures on \mathcal {D}_0^\mathrm {fop}, and fiberwise opposites lifts to an involution of \mathsf {Alg}(\overline {(-)}|_\mathrm {det}).

Let \mathcal {C} be a pullback-positive Markov category. The adjunction \overline {(-)} \dashv (-)|_\mathrm {det} induces a monad on \mathsf {Fib}(\mathcal {C}_\mathrm {det}). A module for this monad is called a stochastic module over \mathcal {C} (or, to distinguish it from the copresheaves of Definition [efr-Y926], a stochastic module fibration). The category of stochastic module fibrations is denoted \mathsf {SFib}(\mathcal {C})

Let \mathcal {D} be a stochastic module over \mathcal {C}, and let

In particular, this operation depends only on s. Moreover, it is functorial, in the sense that given a diagram

The operations \alpha ^* associated to stochastic lifts are "functorial" in the sense that (\alpha \beta )^* = \beta ^*\alpha ^*. However they are not functorial in the sense that \alpha ^*(fg) = \alpha ^*(f)\alpha ^*(g)

To make sense of this, consider a simple case of a map m: I \to X in Kl(\Delta ). Given two objects over * (in Kl(\Delta )^\to ), a map \bar {A}_X \to \bar {B}_X is equivalent to a parametrized map X \times \bar {A} \to \bar {B}. The operation m^* consists in sampling this parameter according to the distribution m---but since composition in the fiber over X is defined by copying the parameter, but composition in the fiber over * (i.e just Kl(\Delta )) is defined by composing the kernels under conditional independence, these only agree if the distribution m is assumed to be deterministic.

Note that if either f or g is pulled back from a map \bar {A} \to \bar {B} (i.e, if they do not depend on the parameter X), the composition is preserved.

Let p: \mathcal {D} \to \mathcal {C} be a Markov prefibration. We say p (or, abusing notation, \mathcal {D}) admits weak conditionals if, given a Cartesian map \bar {Y} \to \bar {Z} and any map \bar {X} \to \bar {Z}, the existence part of the Cartesian condition holds---that is, every factorization p(\bar {X}) \to p(\bar {Y}) admits a lift, although not necessarily a unique one.

We say a Markov category \mathcal {C} admits weak conditionals if its codomain functor \mathcal {C}^\to \to \mathcal {C} 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 \mathcal {C}_\mathrm {det} \to \mathcal {C} (in other words, that they satisfy the existence part of the universal property even for pairs of nondeterministic maps).

Suppose \mathcal {C} admits weak conditionals, and let \mathcal {D} \to \mathcal {C}_\mathrm {det} be a fibration. Then two morphisms f_0,f_1: \bar {X} \to \bar {Y} in \overline {\mathcal {D}}, represented by commutative diagrams

Consider the adjunction |-|: \mathsf {Mon} \leftrightarrows \mathsf {Set} : (-)^* between the category of monoids and the category of sets. Given a monoid M, |M|^* consists of lists of elements in M, and ||M|^*|^* consists of lists of such lists. The two maps ||M|^*|^* \to |M|^* consist in either concatenating the lists, or replacing each list with its product. Clearly these two maps are coequalized by the product map |M|^* \to M. 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).

Let \mathsf {GrpTop} be the category of topological groups, and let R: \mathsf {GrpTop} \to \mathsf {Set} 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 RL for now. The canonical comparison functor \mathsf {GrpTop} \to \mathsf {Grp} just forgets the topology. Given a (non-topological) group, described by a map RLG \to G, we can form the diagram of topological groups LRLG \rightrightarrows LG. Here LG is the free, discrete group on G and LRLG is the free discrete group on the underlying set of LG. Their coequalizer is simply G equipped with the discrete topology, which is indeed the left adjoint to \mathsf {GrpTop} \to \mathsf {Grp}

If \mathcal {D} is a Markov prefibration, we call it a Markov fibration if the diagram \overline {\operatorname {Free}(\mathcal {D}|_\mathrm {det})} \rightrightarrows \overline {\mathcal {D}|_\mathrm {det}} \to \mathcal {D} is a coequalizer in \mathsf {Cat}_{/\mathcal {C}}.

Given an algebra \alpha : \operatorname {Free}(\mathcal {D}_0) \to \mathcal {D}_0 of the free Markov prefibration monad, let \overline {\operatorname {Free}(\mathcal {D}_0)} \rightrightarrows \overline {\mathcal {D}_0} \in \mathsf {MarkPreFib} be as above. We say \alpha presents a Markov fibration if the coequalizer of these maps in \mathsf {Cat}_{/\mathcal {C}} is a Markov prefibration.

Let \alpha : \operatorname {Free}(\mathcal {D}_0) \to \mathcal {D}_0 be stochastic module which presents a Markov fibration. Then the underlying algebra of the Markov prefibration obtained as the coequalizer of \overline {\operatorname {Free}(\mathcal {D}_0)} \rightrightarrows \bar {\mathcal {D}_0} is isomorphic to \alpha , and in particular this prefibration is a Markov fibration.

This correspondence determines an equivalence of categories between the full subcategory \mathsf {MarkFib}(\mathcal {C}) of \mathsf {MarkPreFib}(\mathcal {C}) spanned by the Markov fibrations, and the full subcategory \mathsf {SFib}(\mathcal {C})^p \subseteq \mathsf {SFib}(\mathcal {C}) spanned by those algebras which present a Markov fibration.

Let \mathcal {D}_0 \to \mathcal {C}_\mathrm {det} be a stochastic module fibration.

Let \alpha : X \to M be a stochastic section, let \bar {X} \in \mathcal {D}_{0,X} be an object, and let \phi : \bar {X}_M \to \bar {X}_M \in \mathcal {D}_{0,M} be an endomorphism of its pullback. Observe that if there exists f: N \to M so that f^*(\phi ) = 1 and \alpha factors over f, then \alpha _*(\phi ) = 1. We say \mathcal {D}_0 has weak supports if this implication is an equivalence

Let F,G: \mathcal {D} \rightrightarrows \mathcal {D}' be a parallel pair in \mathsf {Cat}_{/\mathcal {C}}, and suppose both are identity on objects. Suppose moreover this is a reflexive pair, i.e there is some S: \mathcal {D}' \to \mathcal {D} so that FS = GS = 1_{\mathcal {D}'}. Then the coequalizer in \mathsf {Cat}_{/\mathcal {C}} is again identity on objects, and is given on hom-sets simply by the coequalizer of the parallel pair \mathcal {D}(x,y) \rightrightarrows \mathcal {D}'(x,y)

Let \mathcal {D}_0 be a fibration equipped with a stochastic module structure. Consider the equivalence relation on \overline {\mathcal {D}_0}(\bar {X},\bar {Y}) which identifies two precharts (M, \phi ), (N, \phi ') if there exists a map f: N \to M over X,Y and a stochastic section s of f which preserves the section from X, so that s^*\phi ' = \phi (note that this makes sense because pullbacks compose).

1. This equivalence relation respects composition, and so defines a category which we denote \mathsf {SChart}(\mathcal {D}_0)
2. In \mathsf {Cat}_{/\mathcal {C}}, \overline {\operatorname {Free}(\mathcal {D}_0)} \rightrightarrows \overline {\mathcal {D}_0} \to \mathsf {SChart}(\mathcal {D}_0) is a coequalizer diagram. In particular, \mathcal {D}_0 presents a Markov fibration if and only if \mathsf {SChart}(\mathcal {D}_0) is a Markov prefibration (in which case \mathsf {SChart}(\mathcal {D}_0) is the fibration it presents)
3. If \mathcal {D}_0 has weak supports, \mathsf {SChart}(\mathcal {D}_0) is a prefibration

Proof

• April 5, 2025
• Eigil Fjeldgren Rischel

Recall that the fibration \operatorname {Free}(\mathcal {D}_0) has fibers whose morphisms \bar {X} \to \bar {X}' are given by tuples s: X \to M: p, p^*\bar {X} \to p^*\bar {X}' (up to a certain equivalence relation). Forming the free Markov prefibration \overline {\operatorname {Free}(\mathcal {D}_0)} on this fibration, we find that the morphisms are given by diagrams

equipped with a map \phi : p^*q^*\bar {X} \to p^*r^*\bar {Y}. The two maps \overline {\operatorname {Free}(\mathcal {D}_0)} \to \overline {\mathcal {D}_0} carry such a thing to first, the map resulting from forgetting M and just composing q,r with p to get a span (and composing the sections to get a new section), and secondly, the map with apex M obtained by using the stochastic module structure to push \phi down into a map over M. It is clear that this is equivalently the equation described in the theorem. This establishes points 1. and 2., since by Lemma [efr-L7L2] we can compute such coequalizers hom-set by hom-set.

Now we wish to prove that \mathsf {SChart}(\mathcal {D}_0) is a Markov prefibration given weak supports. Since by the coequalizer presentation, its deterministic part is isomorphic to \mathcal {D}_0, the fibration property is automatic. It remains to verify that, given a triangle

in \mathcal {C} with the vertical and horizontal maps deterministic, an object \bar {A} \in \mathcal {D}_{0,Y} and Cartesian maps \bar {A}_X \to \bar {A} \leftarrow \bar {A}_Z, there exists a unique lift \bar {A}_X \to \bar {A}_Z in \mathsf {SChart}(\mathcal {D}_0).

Such a lift is given by a diagram

equipped with p^*\bar {A}_X \to q^*\bar {A}_Z. By taking the equalizer of the two maps M \to Y (the section factors over this), we may assume these two are equal, which implies that the pulled-back objects are equal---denote this object \bar {A}_M. Now the hypothesis is that after postcomposing with the Cartesian map \bar {A}_Z \to \bar {A}, this gives the map \bar {A}_X \to \bar {A}. This postcomposition is given simply by postcomposing the leg M \to Z with the map Z \to Y (and observing that, by functoriality of pullbacks, this does not alter the pulled-back object). Then the claim is that integrating this map \bar {A}_M \to \bar {A}_M down into a map \bar {A}_X \to \bar {A}_X, it gives the identity. But by assumption this means we can pull back to some object M' \to M (lifting the section from X) where the two maps are already equal to the identity. But this pull-back can be applied to the original map \bar {A}_X \to \bar {A}_Z as well. But this implies every such map is equal to the one represented by the diagram

and the identity on \bar {A}_{X \otimes Y}, with the map M' \to X \otimes Y giving the witness, since identities pull back. This map only depends on the underlying X \to Y, hence \mathsf {SChart}(\mathcal {D}_0) is indeed a prefibration.

Let (M,\phi ) and (N,\psi ) be two representatives of charts. Given some possibly stochastic map f: M \to N over X and Y, recall (Lemma [efr-VF6V]) that we can define f^*\psi , regardless of whether f is deterministic or the section of a deterministic map. If there exists such a map f, we can always factor it over the pullback M \times _{X \times Y} N as a section followed by a deterministic map. Hence the equivalence relation defining \mathsf {SChart} is equivalent to the relation identifying two representatives whenever there exists such an f with f^*\psi = \phi

By construction, for each Markov prefibration \mathcal {D}, there is a canonical functor \mathsf {SChart}(\mathcal {D}|_\mathrm {det}) \to \mathcal {D} over \mathcal {C}, which restricts to an isomorphism on the deterministic part (and in particular preserves Cartesian morphisms). \mathcal {D} is a Markov fibration if and only if this is an isomorphism.

Also by construction, given a morphism of stochastic modules F: \mathcal {D}_0 \to \mathcal {D}_0', there is an induced functor \mathsf {SChart}(\mathcal {D}) \to \mathsf {SChart}(\mathcal {D}') over \mathcal {C}. This restricts to F on the deterministic part and in particular preserves Cartesian morphisms.

Let \mathcal {D}_0 be a stochastic module fibration. Then \mathcal {D}_0 presents a Markov fibration if and only if \mathcal {D}_0^\mathrm {fop} does it.

Proof

• April 27, 2025
• Eigil Fjeldgren Rischel

Consider a triangle of this for in \mathcal {C}:

where the maps to Z are deterministic. Suppose given Cartesian lifts \bar {X} \to \bar {Z}, \bar {Y} \to \bar {Z} of the cospan. These are the same in both cases, coming from Cartesian maps in \mathcal {D}_0 \mathcal {D}_0^\mathrm {fop} (which are the same). We must show that there is a unique lift of f to a chart in \mathsf {SChart}(\mathcal {D}_0) if and only if there is a unique lift to a chart in \mathsf {SChart}(\mathcal {D}_0^\mathrm {fop}). Clearly it suffices to prove the "only if" implication, so suppose \mathsf {SChart}(\mathcal {D}_0) is a prefibration.

By passing to the equalizer as in the proof that \overline {\mathcal {D}_0} is a prefibration, we may assume that any such lift is represented by a diagram

where the outer square and the triangle X \to M \to Y commutes, and f' is a section.

Take such a diagram and let \phi : \bar {Z}_M \to \bar {Z}_M be the map representing a chart. Then the claim is there exists some zig-zag of chart equivalences identifying (M,\phi ) with (M,1). But clearly this is invariant under passing to the fiberwise opposite, and so \mathcal {D}^\mathrm {fop} is also a prefibration.

We refer to \mathsf {SChart}(\mathcal {D}_0) as the category of stochastic charts in \mathcal {D}_0. We refer to \mathsf {SChart}(\mathcal {D}_0^\mathrm {fop}) as stochastic lenses and denote it also \mathsf {SLens}(\mathcal {D}_0)

Kl(\Delta )^\to 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 {Kl(\Delta )^\to |_\mathrm {det}}, given by a diagram

We can factor the section X \to M as X \to X \times Y \to M, where the first map is just the pairing and the second is a conditional distribution. This induces a factorization of the lift \bar {X} \to M \times _X \bar {X} over \bar {X} \to \bar {X} \times Y. By composing the map M \times _X \bar {X} \to \bar {Y} with this factorization to build the map \bar {X} \times Y \to \bar {Y}, we have found a new representative for the same map.

Hence every map over X \to Y has a representative where the residual is X \times Y. We would like to argue that, since the map \bar {X} \times Y \to \bar {Y} is given by the conditional distribution of the composite map \bar {X} \to \bar {Y}, it is uniquely determined by it, and thus if two distinct maps in \overline {Kl(\Delta )^\to |_\mathrm {det}} have the same underlying map in Kl(\Delta )^\to , they must have equal representatives of this form, and so be identified in the coreflection (which must therefore be isomorphic to Kl(\Delta )^\to ). But of course, the two maps may only be almost certainly equal.

In this case, there is a simple fix: instead of taking X \times Y as the residual, take the subset S given by those pairs (x,y) where y has positive probability given x. The pairing factors over this, of course, and two maps \bar {X} \times _X S \to \bar {Y} which give the same distribution \bar {X} \to \bar {Y} really must have the same value on every point. This proves that \mathsf {SChart}(Kl(\Delta )^\to ) \to (Kl(\Delta ))^\to is faithful and hence an isomorphism.

\mathsf {BorelStoch}^\to , as we have noted, is a Markov prefibration, and hence induces a stochastic module structure on \mathsf {BorelStoch}^\to |_\mathrm {det}. 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, \mathsf {SLens}(\mathsf {BorelStoch}^\to |_\mathrm {det}), would be a prefibration whose deterministic part was \mathsf {BorelStoch}^\to |_\mathrm {det}^\mathrm {fop}. But Example [efr-8B5X] shows that this is impossible.

Let \mathcal {C} be a Markov category with supports. Then the stochastic module induced by \mathcal {C}^\to \to \mathcal {C} presents a Markov fibration. If \mathcal {C} has conditionals, this Markov fibration is isomorphic to \mathcal {C}^\to .

Proof

• April 6, 2025
• Eigil Fjeldgren Rischel

Given a section s: X \to M:p and \phi : A \times _X M \to A \times _X A, simply take the pullback to the support of s. It must be the case that \phi (a,m) = (a,m) s-almost surely, which implies strict equality on the support. Hence by Proposition [efr-TBZZ], \mathcal {C}^\to presents a Markov fibration.

There is an induced map \mathsf {SChart}(\mathcal {C}^\to |_\mathrm {det}) \to \mathcal {C}^\to , which we claim is an isomorphism. So consider a map in \mathcal {C}^\to :

This map is in the image of

where the map \bar {X} \times Y \to \bar {Y} is taken to be a conditional. Note that every map in \mathsf {SChart}(\mathcal {C}^\to |_\mathrm {det}) can be represented in this form, by taking a conditional of M given Y to build a section to M \to X \otimes Y. Since conditionals are almost-surely equal, by restricting to the support of X \to X \otimes Y, we can find a representative which only depends on the overall map \bar {X} \to \bar {Y}, which proves that the map from \mathsf {SChart} is faithful, concluding the proof.

If \mathcal {C} is Cartesian (even if it does not admit pullbacks), the definition of Markov fibration over \mathcal {C} still makes sense, the Markov fibrations are exactly the Grothendieck fibrations, and their fiberwise opposites are simply their fiberwise opposites in the usual sense.

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:

1. If we let \mathcal {C}_{/a}^\mathrm {det} refer to the full subcategory of the slice spanned by the deterministic morphism x \to a, we have an equivalence of categories \mathcal {C}_{/a}^\mathrm {det} \times \mathcal {C}_{/b}^\mathrm {det} \cong \mathcal {C}_{/a + b}^\mathrm {det}, given by taking coproducts
2. \mathcal {C}_\mathrm {det} is an extensive category in the usual sense and the inclusion \mathcal {C}_\mathrm {det} \to \mathcal {C} preserves pullbacks along coproduct inclusions.

More generally, for an infinite regular cardinal \kappa , we say that \mathcal {C} is \kappa -extensive if \mathcal {C}_\mathrm {det} is a \kappa -extensive category in the ordinary sense and both coproduct inclusions and pullbacks along them are preserved by the functor \mathcal {C}_\mathrm {det} \to \mathcal {C}.

Let \mathcal {C} be an extensive Markov category, let \mathcal {D}_0 \to \mathcal {C}_\mathrm {det} be a fibration which satisfies \mathcal {D}_{0,X+Y} = \mathcal {D}_{0,X} \times \mathcal {D}_{0,Y}. Note that this implies \mathcal {D}_0 admits finite coproducts, and they're given exactly by this pairing. Suppose \mathcal {D}_0 is equipped with a stochastic module structure. Then \mathcal {D}_0 \hookrightarrow \mathsf {SChart}(\mathcal {D}_0) preserves the finite coproducts. In particular, \mathcal {D}_0^\mathrm {fop} has the same coproducts as \mathcal {D}_0, and \mathcal {D}_0^\mathrm {fop} \to \mathsf {SLens}(\mathcal {D}_0) preserves them as well.

Suppose \mathcal {D} is a Markov fibration so that each pullback functor f^*: \mathcal {D}_Y \to \mathcal {D}_X for f: X \to Y \in \mathcal {C}_\mathrm {det} can be taken to be bijective on objects, and that these can furthermore be chosen strictly functorial (so that (fg)^* = g^*f^*). Then, writing objects \bar {X} \in \mathcal {D}_X as \binom {A \in \mathcal {D}_*}{X \in \mathcal {C}}, where A is the unique object in \mathcal {D}_* which pulls back to \bar {X} under the deletion X \to *, (note that this means f^*\binom {A}{Y} = \binom {A}{X}) we may characterize the fiberwise dual as having hom-sets \mathcal {D}^{\mathrm {fop}}(\binom {A}{X},\binom {B}{Y}) = \mathcal {D}(\binom {B}{X},\binom {A}{Y})

Let \mathcal {D} \to \mathcal {C} be a Markov prefibration, and let F: \mathcal {C}' \to \mathcal {C} be any functor from another Markov category which preserves deterministic maps. Then the pullback \mathcal {D}' = \mathcal {D} \times _{\mathcal {C}} \mathcal {C}' \to \mathcal {C}' is again a Markov prefibration, and the functor \mathcal {D}' \to \mathcal {D} preserves Cartesian maps.

Proof

• April 9, 2025
• Eigil Fjeldgren Rischel

Since pullbacks compose, \mathcal {D}'|_\mathrm {det} \to \mathcal {C}'_\mathrm {det} is the pullback of \mathcal {D}|_\mathrm {det} along \mathcal {C}'_\mathrm {det} \to \mathcal {C}_\mathrm {det}. Since fibrations are stable under pullback, this is a fibration.

Now consider a triangle in \mathcal {C}':

with X, A \to B deterministic, and let \bar {B}_X \to \bar {B} \leftarrow \bar {B}_A be Cartesian maps lying over these. We must show there is a unique lift \bar {f}: \bar {B}_X \to \bar {B}_A rendering the lifted triangle commutative. By definition, to give such a morphism is to given one over F(f), and the triangle commutes if and only if its image in \mathcal {D} commutes.

The triangle in \mathcal {C}' goes to a triangle of the same class in \mathcal {C}, and the Cartesian maps go to Cartesian maps of the same type. Hence there is a unique lift of F(f) of the given type, which is exactly what we needed to show.

Given any functor F: \mathcal {C} \to \mathcal {C}' between Markov categories, we may attempt to define an oplax monoidal structure F(X \otimes Y) \to F(X) \otimes F(Y) by pairing the projections.

This is not necessarily a natural transformation. However, if it is, F it automatically equips F 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 [fritz-synthetic-markov-cats] 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.)

We will denote by \mathsf {MarkPreFib} the category whose objects are Markov prefibrations \mathcal {D} \to \mathcal {C}, and whose functors are commutative squares

We will let \mathsf {Markov}^\mathrm {oplax} denote the category of Markov categories and oplax Markov functors. Note that by Lemma [efr-9VAH], the forgetful functor \mathsf {MarkPreFib} \to \mathsf {Markov}^\mathrm {oplax} is a fibration.

Let \mathsf {Markov}_s \subseteq \mathsf {Markov}^\mathrm {oplax} denote the subcategory of strictly monoidal Markov functors (i.e those where the oplaxator F(X \otimes Y) \to F(X) \otimes F(Y) is the identity).

1. \mathsf {Markov}_s and \mathsf {Markov}^\mathrm {oplax} admit all products, computed simply as products in \mathsf {Cat}.
2. \mathsf {Markov}_s admits all finite limits, and these are preserved by the inclusion into \mathsf {Markov}^\mathrm {oplax}

The category \mathsf {MarkPreFib}(\mathcal {C}) admits all products, and pullbacks along isofibrations. These are simply computed as limits in \mathsf {Cat}_{/\mathcal {C}}

The functor \overline {(-)}: \mathsf {Fib} \times _\mathsf {Cat} \mathsf {Markov}^\mathrm {oplax} \to \mathsf {MarkPreFib} preserves products.

For each Markov category \mathcal {C} with weak conditionals, \overline {(-)}: \mathsf {Fib}(\mathcal {C}_\mathrm {det}) \to \mathsf {MarkPreFib}(\mathcal {C}) preserves the terminal object, and pullbacks along isofibrations.

For any (pullback-positive) \mathcal {C}, the monad \overline {(-)}|_\mathrm {det} on \mathsf {Fib}(\mathcal {C}_\mathrm {det}) preserves products.

The property of having weak supports is stable under equalizers in stochastic module fibrations over \mathcal {C}. If \mathcal {C} has weak conditionals, it is also stable under finite products (hence all finite limits).

A monoidal Markov prefibration is a markov prefibration p: \mathcal {D} \to \mathcal {C} equipped with a monoidal category structure on \mathcal {D} so that p is strict monoidal and so that the underlying fibration is a monoidal fibration (i.e so that p preserves Cartesian lifts).

A braided or symmetric monoidal Markov prefibration is a monoidal prefibration equipped with a braiding or symmetry on \mathcal {D} so that p is moreover a braided monoidal functor.

Let F,G: \mathcal {C}' \rightrightarrows \mathcal {C}, S: \mathcal {C} \to \mathcal {C}' be a reflexive pair of identity-on-objects, strict monoidal functors. Let E: \mathcal {C} \to \mathcal {D} be the coequalizer in \mathsf {Cat}. Then \mathcal {D} inherits a monoidal structure making E strict monoidal. Moreover, if \mathcal {C} is symmetric or braided, \mathcal {D} inherits this structure making E a braided functor.

1. The free prefibration monad on \mathsf {Fib}(\mathcal {C}_\mathrm {det}) has a canonical lifting to \mathsf {MonFib}(\mathcal {C}_\mathrm {det})
2. Given a monoidal prefibration, its underlying stochastic module acquires the structure of an algebra of this lifted monad.
3. Given an algebra for the lifted monad \mathcal {D}_0, \mathsf {SChart}(\mathcal {D}_0) acquires a monoidal structure so that \mathsf {SChart}(\mathcal {D}_0) \to \mathcal {C} is a strict monoidal functor.
4. If \mathcal {D}_0 moreover has weak supports, this forgetful functor is a monoidal prefibration.

Every statement holds also for braided or symmetric fibrations.

Proof

• April 25, 2025
• Eigil Fjeldgren Rischel

By Reference [moeller-vasilakopoulou], \mathsf {MonFib}(\mathcal {C}_\mathrm {det}) is equivalent to the category of pseudomonoids in \mathsf {Fib}(\mathcal {C}_det). Since the monad preserves products, it must preserve pseudomonoids, which is all we need.

Now suppose \mathcal {D} \to \mathcal {C} is a monoidal prefibration. Then its underlying fibration is a monoidal fibration, hence an object of \mathsf {MonFib}(\mathcal {C}_\mathrm {det}). The claim is that the functor \overline {\mathcal {D}|_\mathrm {det}}|_\mathrm {det} \to \mathcal {D}|_\mathrm {det} is monoidal. The induced monoidal structure is given on objects by \otimes _\mathcal {D} and takes a pair of morphisms in the fiber represented by sections (s: X \to M, \phi : \bar {X_0}_M) \to \bar {X_1}_M and s': X \to M', \phi ': \bar {X'_0}_M \to \bar {X'_1}_M to \langle s,s' \rangle : X \to M \times _X M', \pi _M^*\phi \otimes _\mathcal {D} \pi _{M'}^*\phi '. Recalling that the algebra structure is defined by taking (s,\phi ) to the composite \bar {X_0} \to \bar {X_0}_M \xrightarrow {\phi } \bar {X_1}_M \to \bar {X_1}, and chasing the below diagram around, it is apparent that the algebra structure preserves the monoidal structure.

Now let \mathcal {D}_0 be a monoidal stochastic module in this sense. It suffices to show that the free prefibration \overline {\mathcal {D}_0} is a monoidal prefibration, by Lemma [efr-7GH5], and the induced functor \mathsf {SChart}(\mathcal {D}_0) \to \mathcal {C} will clearly be strict monoidal if \overline {\mathcal {D}_0} \to \mathcal {C} is.

To construct this monoidal structure on \mathcal {D}_0, simply note that since \overline {(-)} preserves global limits as well, there is an induced monoidal structure on \overline {\mathcal {D}_0} so that the forgetful functor is strict monoidal. Recalling that the Cartesian lifts of f: X \to Y \in \mathcal {C} to \overline {\mathcal {D}_0} are given by the span X = X \to Y and the morphism 1_{f^*\bar {Y}}, it is easy to see by unwinding the definition that these are stable under tensor.

If \mathcal {D}_0 has weak supports, we have already proven that \mathsf {SChart}(\mathcal {D}_0) \to \mathcal {C} 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 \overline {\mathcal {D}_0}, the preceding claim that they are stable under tensor implies the same for \mathsf {SChart}(\mathcal {D}_0), finishing the proof.

The last point is mostly trivial. The product preservation still establishes the lifting to \mathsf {BrMonFib}(\mathcal {C}_\mathrm {det}) and \mathsf {SymMonFib}(\mathcal {C}_\mathrm {det}). 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 \mathsf {SChart} follows again from Lemma [efr-7GH5], and there is nothing to show for the last point.

A monoidal Markov fibration is a monoidal prefibration \mathcal {D} \to \mathcal {C} which is a Markov fibration. It is braided or symmetric if it is braided or symmetric as a prefibration

Since \overline {(-)} preserves both global products and fiberwise ones, it induces both a fiberwise monoidal structure and a "global" monoidal structure on \overline {\mathcal {D}_0}. Here we only use the global one. If \mathcal {C} were Cartesian, the global one would be induced from the local one by, given maps \bar {X_1} \to \bar {X_2}, \bar {Y_1} \to \bar {Y_2}, pulling each of them back along the square

Let \mathcal {D} be a monoidal stochastic module fibration over \mathcal {C} and suppose each fiber \mathcal {D}_X is a Markov category, and this structure is preserved by the pullback functors f^*. Then \mathsf {SChart}(\mathcal {D}) carries the structure of a Markov category, so that \mathsf {SChart}(\mathcal {D}) \to \mathcal {C} is a Markov functor.

If \mathcal {D} \to \mathcal {C} is a Markov prefibration which is also a strict Markov functor, the induced monoidal structure on \mathcal {D}|_\mathrm {det} acquires a fiberwise Markov structure.

Let \mathcal {D} be a stochastic module fibration, and suppose each fiber has coproducts, and these are preserved by the pullback functors. Then \mathsf {SLens}(\mathcal {D}) is a Markov category, with monoidal structure given by \binom {\bar {X}}{X} \& \binom {\bar {Y}}{Y} = \binom {\pi _X^*\bar {X} + \pi _Y^*\bar {Y}}{X \otimes Y}

Let \mathcal {C} act on \mathcal {D}. Then \mathsf {Optic}_\mathcal {C}(\mathcal {D}) := \mathsf {Optic}_\mathcal {C}(\mathcal {C},\mathcal {D}) has a functor to \mathcal {C}. The deterministic part \mathsf {Optic}_\mathcal {C}(\mathcal {D})|_\mathrm {det} admits the structure of a stochastic module fibration. There is an isomorphism \mathsf {Optic}_\mathcal {C}(\mathcal {D}) \to \mathsf {SChart}(\mathsf {Optic}_\mathcal {C}(\mathcal {D})|_\mathrm {det}).

If \mathcal {D} is itself symmetric monoidal and the action is symmetric (meaning it is given by M \cdot A = F(M) \otimes A for some symmetric monoidal functor F: \mathcal {C} \to \mathcal {D}, see Reference [actegories-amthematician-capucci-gavranovic] 5.4.3 and 5.5.12), this stochastic module is symmetric monoidal and the isomorphism is an isomorphism of symmetric monoidal categories.

Let \mathcal {C} be any pullback-positive Markov category. Then \mathcal {C}^\to \to \mathcal {C} is a Markov prefibration which thus induces a stochastic module structure on \mathcal {C}^\to |_\mathrm {det}. Writing simply \mathsf {SChart}(\mathcal {C}), \mathsf {SLens}(\mathcal {C}) for \mathsf {SChart}(\mathcal {C}^\to |_\mathrm {det}), \mathsf {SLens}(\mathcal {C}^\to |_\mathrm {det}), we have:

1. There is a functor \mathsf {Optic}(\mathcal {C}) \to \mathsf {SLens}(\mathcal {C}), which is fully faithful. Dually there is a functor \mathsf {coOptic}(\mathcal {C}) \to \mathsf {SChart}(\mathcal {C}) which is fully faithful.
2. \mathsf {SLens}(\mathcal {C}) and \mathsf {SChart}(\mathcal {C}) both admit symmetric monoidal structures, which make the functors \mathsf {SChart}(\mathcal {C}), \mathsf {SLens}(\mathcal {C}) \to \mathcal {C} strict symmetric monoidal, as well as the functors \mathsf {Optic}(\mathcal {C}) \to \mathsf {SLens}(\mathcal {C}), \mathsf {coOptic}(\mathcal {C}) \to \mathsf {SChart}(\mathcal {C}) strong symmetric monoidal.
3. If \mathcal {C} is extensive, this functor preserves the coproducts {A \choose X} + {A \choose Y} = {A \choose X+Y}, and \mathsf {SChart}(\mathcal {C}),\mathsf {SLens}(\mathcal {C}) both admit all finite coproducts.
4. If \mathcal {C} moreover has conditionals and supports, \mathsf {SChart}(\mathcal {C}) = \mathcal {C}^\to

Given a compact Hausdorff space X, a Banach space bundle is a space over X, V \to X, equipped with a fiberwise (complex) vector space structure +: V \times _X V \to V, \cdot : \mathbb {C} \times V \to V, so that there exists a cover \{U_i\} of X so that for each U_i, there exists a Banach space V_i and a homeomorphism V \times _X U_i =: V_{U_i} \cong V_i \times U_i over U_i, which is moreover linear in each fiber, where V_i is equipped with the norm topology.

Note that this determines a local norm on each V_{U_i} (and in particular each V_x) up to equivalence (but no stricter than that). In particular each V_x is a Banach space.

A morphism of Banach space bundles is a continuous map f: V \to W over X which is linear on each bundle. Note that this implies that on a suitable cover U_i, the maps f: V_{U_i} \to W_{U_i} obey \left \lVert f(v) \right \rVert \leq C_i \left \lVert v \right \rVert for some C_i \in \mathbb {R}, for any local norms inducing the topologies, and hence by compactness there exists some C so that \left \lVert f(v) \right \rVert \leq C \left \lVert v \right \rVert for each v.

If X \to Y is a continuous map, there is a pullback functor \mathsf {Ban}_Y \to \mathsf {Ban}_X. The Grothendieck construction of this gives a fibration \mathsf {BanBun} \to \mathsf {CHaus}

Note that Tychonoff spaces include all compact Hausdorff spaces. Therfore consider the full subcategory \mathsf {CHausStoch} \hookrightarrow \mathsf {TychStoch} spanned by these. The fibration \mathsf {BanBun} admits the structure of a stochastic module: given M \to X, s: X \to M a kernel, and a linear continuous map f: V \times _X M \to W \times _X M, given v \in V_x, there is an induced function M_x \to W_x given by f(v,-). Since this is bounded (being continuous on a compact space) it is (Bochner) integrable, define s^*f(v) 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 [efr-A6YR]). 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 M \times \mathbb {R}^n \to \mathbb {R}^n and a measure on M, the fact that the expectation map \mathbb {R}^n \to \mathbb {R}^n is the identity by no means implies that the original map is almost surely the identity or anything like that. If (m,e_0) \mapsto e_1 and m has positive probability, this can be canceled out by (m',e_0) \mapsto -e_1. 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.

Let \mathcal {C} be a representable, positive Markov category so that \mathcal {C}_\mathrm {det} admits intersections and the probability monad P preserves them. Then each slice (\mathcal {C}_\mathrm {det})_{/X} inherits a monad structure given by P_X(B \to X) = P(B) \times _{P(X)} X. This is pseudofunctorial in X. Moreover, the stochastic module structure on \mathcal {C}^\to |_\mathrm {det} extends to a stochastic module structure on the fibration representing the pseudofunctor X \mapsto \mathsf {Alg}(P_X).

Proof

• May 2, 2025
• Eigil Fjeldgren Rischel

The monad is induced by the adjunction \mathcal {C}_\mathrm {det}{/X} \leftrightarrows \mathsf {Alg}(P)_{PX}. Let f: X \to Y (deterministic). For abstract reasons there is a natural transformation P_Yf^* \to f^*P_X. Writing this out, we find P(A \times _Y X) \times _{PX} X \to P(A) \times _{PY} X

By representability, the unit X \to PX is a monomorphism. Hence a map into P(A \times _Y X) \times _{PX} X is precisely a map in \mathcal {C} into A \times _Y X so that the marginal on X is deterministic. But by pullback-positivity this is precisely a map (in \mathcal {C}_\mathrm {det}) into P(A) \times _{PY} X.

Let M \to X, two P_X-algebras A, B, and a map M \times _X A \to M \times _X B which is a homomorphism for the induced P_M-algebras, and a map X \to PM be given. Note that by the above, P_M(M \times _X A) \cong M \times _X P_XA. Then the induced map A \to B is given by A \to PM \times _{PX} A \cong P_X(M) \times _X A \hookrightarrow P_X(M \times _X A) \to P_X(B) \to B

We must show this is a P_X-homomorphism. Let us simplify by working internally to (\mathcal {C}_{\mathrm {det}})_{/X}---thus we have a Cartesian category equipped with a strong commutative monad P, a map * \to PM, and a map M \times A \to B which is a parametrized algebra homomorphism, in the sense that the diagram

commutes. Now we must show the map A \to PM \times A \to P(M \times A) \to PB \to B is a P-homomorphism. Write E_A,E_B for the structure maps of the two algebras. Consider this diagram:

The triangle at the top left commutes because P is strong. The square to the right does not commute in general---however, since P is commutative, the composite maps PM \times PA \to P^2(M \times A) \xrightarrow {\mu } P(X \times A) agree. Since the map P^2(M \times A) \to B factors over this, we may replace one edge of this square with another. The square to the right of that is simply P(-) 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 P is strong. Hence the outer square commutes, which is precisely the homomorphism property we wanted.

It is apparent that, if A, B = P_XA', P_B' are free algebras, this restricts to the stochastic module structure of \mathcal {C}^\to |_\mathrm {det} (viewing \mathcal {C} as the Kleisli category of P). 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.

Let \mathcal {M} be a monoidal category, and let \bullet : \mathcal {M} \times \mathcal {C} \to \mathcal {C} be an action of \mathcal {M} on another category \mathcal {C}. Then the \mathsf {Para} construction is the category \mathsf {Para}_\mathcal {M}(\mathcal {C}) where

1. Objects are objects of \mathcal {C}
2. Morphisms A \to B are tuples P \in \mathcal {M}, P \bullet A \to B up to the natural notion of isomorphism.
3. Composition is by tensoring the parameter objects and composing.

Let \mathcal {C} be a monoidal category, and let S: \mathcal {C} \to \mathsf {Set} be any functor. Recall that \int S is the category whose objects are pairs (X \in \mathcal {C}, s \in S(X)) and whose morphisms (X,s) \to (Y,s') are f: X \to Y so that S(f)(s) = s'. If S is lax monoidal (for (\mathsf {Set},\times ),) \int S acquires a lax monoidal structure making the forgetful functor \int S \to \mathcal {C} strong (even strict) monoidal.

With the action induced by this functor, the bicategory \mathsf {Para}_{\int S}(\mathcal {C}) has morphisms given by maps M \otimes X \to Y, m \in S(M), and maps given by reparametrizations which preserve the decoration m.

For example, let \mathcal {C} be a Markov category and let S(X) = \mathcal {C}(I,X). Then morphisms are parametrized maps equipped with a measure on the parameter space.

If \mathbb {C} = \mathsf {Set} regarded as a discrete 2-category, as noted, a pseudomonoid action is just a monoid action in the ordinary sense---that is, a monoid M, a set X and a function m \cdot x so that m \cdot (n \cdot x) = (mn) \cdot x. The para construction is then the action category, whose objects are the points of X and whose morphisms x \to y are elements m so that m \cdot x = y (with multiplication as composition).

Let \operatorname {\mathsf {Ab} - \mathsf {Cat}} be the 2-category of \mathsf {Ab}-enriched categories, functors and natural transformations. Recall that a ring R is the same as a one-object category enriched in \mathsf {Ab}, and it admits a monoidal structure if and only if it is commutative (this is not particular to \mathsf {Ab}-enriched categories). An enriched R-action on an \mathsf {Ab}-category \mathcal {C} is then an R-module structure on each hom-set so that composition is R-linear in each variable.

If a category has coproducts, then \mathsf {Set} acts on it via S \cdot X = \coprod _{s \in S}X. (This is the \mathsf {Set}-enriched case of what in enriched categories is called a copower or tensor, the dual of the power objects from Example [efr-K3YI]). The morphisms of \mathsf {Para}_\mathsf {Set}(\mathcal {C}) are pairs (I \in \mathsf {Set}, (f_i: X \to Y \in \mathcal {C})_{i \in I}), which compose in the obvious way.

Note that this definition clearly makes sense even if \mathcal {C} does not actually have coproducts. This is an example of another construction which has been called \mathsf {Para}, which takes a monoidal category \mathcal {V} and a \mathcal {V}-enriched category \mathcal {C} and constructs a double category where the morphisms are pairs (J \in \mathcal {V}, J \to \mathcal {C}(X,Y)). 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 [internal-enriched], but these are categories enriched over internal monoidal categories---that is, the ambient category \mathcal {E} is a 1-category and the categorical structure of the base of enrichment \mathcal {V} is formulated on top of this, not as part of the structure of the objects of the category \mathcal {E})

Let \mathcal {M} be a monoidal category, and let \mathcal {C} be a category with an \mathcal {M}-action.

Then there is a pseudo double category \mathsf {\mathbb Para}_\mathcal {M}(\mathcal {C}), with

1. \mathsf {\mathbb Para}_\mathcal {M}(\mathcal {C})_0 = \mathcal {C}
2. \mathsf {\mathbb Para}_\mathcal {M}(\mathcal {C})_1 = \mathcal {M} \times \mathcal {C} \downarrow \mathcal {C}, with domain and codomain given by the two projections to \mathcal {C}
3. The identities map \mathcal {C} \to \mathcal {M} \times \mathcal {C} \downarrow \mathcal {C} is given by (I, 1_\mathcal {C}, 1_\mathcal {C}, \lambda ), where \lambda : I \cdot - \to - is the left unitor of \mathcal {M}.
4. The horizontal composition map is the composition in \mathsf {Para}: given M \cdot X \to Y, N \cdot Y \to Z, their composite is given by (N \otimes M) \cdot X \cong N \cdot (M \cdot X) \to N \cdot Y \to Z. The horizontal composition of 2-cells is defined analogously.

Moreover, if \mathcal {C} \to \mathcal {D} is strict homomorphism of \mathcal {M}-modules, there is an induced pseudofunctor \mathsf {\mathbb Para}_\mathcal {M}(\mathcal {C}) \to \mathsf {\mathbb Para}_\mathcal {M}(\mathcal {D}). If \mathcal {N} \to \mathcal {M} is a strict monoidal functor, then regarding \mathcal {C} as an \mathcal {N}-module along this map, there is an induced pseudofunctor \mathsf {\mathbb Para}_\mathcal {N}(\mathcal {C}) \to \mathsf {\mathbb Para}_\mathcal {M}(\mathcal {C}). These combine into a 2-functor \mathsf {Act}_s \to \mathsf {PsDbl}_s 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.

Proof

• April 29, 2025
• Eigil Fjeldgren Rischel

Note that every actegory \mathcal {M} \curvearrowright \mathcal {C} is equivalent to a strict action of a strict monoidal category \mathcal {M}_s \curvearrowright \mathcal {C}_s (in the sense that there exists \mathcal {M}_s \xrightarrow {\sim } \mathcal {M} strong monoidal equivalence and \mathcal {C}_s \xrightarrow {\sim } \mathcal {C} 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 \mathcal {M} \curvearrowright \mathcal {C}, \mathcal {M}' \curvearrowright \mathcal {C}', F: \mathcal {M} \to \mathcal {M}' be a strict monoidal functor, and let G: \mathcal {C} \to \mathcal {C}' be a linear functor "over F", that is a (strict) \mathcal {M}-linear functor when \mathcal {C}' is viewed as a \mathcal {M}-actegory along F. Then the induced functor \mathsf {\mathbb Para}_\mathcal {M}(\mathcal {C}) \to \mathsf {\mathbb Para}_{\mathcal {M}'}(\mathcal {C}') is given by G on the vertical category, and on the horizontal category by (M,X,X',\phi : M \cdot X \to X') \mapsto (F(M), G(X), G(X'), F(M) \cdot G(X) \simeq G(M \cdot X) \to G(X')) where the isomorphism is the linearity---noting that the M-action on \mathcal {C}' is precisely acting by F(M).

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 H from here.

It suffices to verify this for strict actions. So the composite of M \cdot X \to Y, N \cdot Y \to Z is given by the composite (N \otimes M) \cdot X = N \cdot M \cdot X \to N \cdot Y \to Z.

Given two such composable maps, call them f,g, the two objects H(fg), H(f)H(g) are given by (F(M \otimes N), G(X), G(Z), \beta ), (F(M) \otimes F(N), G(X), G(Z), \alpha ) ,where \alpha , \beta are respectively the maps across the top and bottom of this diagram:

This proves that H 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 \mathsf {PsDbl}_s and \mathsf {Act}_s are computed levelwise, it suffices to observe that the slice category also preserves limits, being a weighted limit itself.

Note that \mathsf {\mathbb Para}_\mathcal {M}(\mathcal {C}) comes equipped with a strict functor to B\mathcal {M} (viewed as a double category). In fact, one can recover the action \cdot : \mathcal {M} \times \mathcal {C} \to \mathcal {C} from this data: the object M \cdot X is characterized up to isomorphism by its universal property: parametrized maps M \cdot X \nrightarrow Y with parameter N are in bijection with maps X \to Y parametrized by N \otimes M.

Let \mathbb {C} be a 2-category, let D: \mathbb {I} \to \mathbb {C} be a 2-diagram in it, and let W: \mathbb {I} \to \mathsf {Cat} be another 2-functor. A limit of D weighted by {W} is an object \lim ^W D \in \mathbb {C} equipped with a natural isomorphism of categories \mathbb {C}(X,\lim ^W D) \cong [ {\mathbb {I} }, {\mathbb {C}} ] (W, \mathbb {C}(X,D(-))) (natural in X \in \mathbb {C}).

If W(-) = * is constant at the point, and \mathbb {I} is an 1-category, then a weighted limit is simply a L \to D(i) in the underlying category \mathbb {C}_0 so that there is additionally an isomorphism of categories \mathbb {C}(X,L) \xrightarrow {\sim } \lim _i \mathbb {C}(X,D(I)). Note that this is stronger than just a limit in \mathbb {C}_0. We will refer to limits of this form by their ordinary names---speaking for example of pullbacks, products, and so on.

If I = *, D(*) = D \in \mathbb {C}, and W(*) = C \in \mathsf {Cat}, the universal property of the weighted limit is that \mathbb {C}(X, \lim ^W D) = \mathsf {Cat}(C, \mathbb {C}(X,D)). In this case we write C \pitchfork D for this limit if it exists, and call it a power of D by C.

If \mathbb {I} = \{A \to B \leftarrow C\} and the weighting is given by W(B) = \{1 \to 2\}, W(A) = \{1\}, W(B) = \{2\} (with the obvious inclusions), then the weighted limit is called the comma object and will be denoted D(A) \downarrow _{D(B)} D(C). Note that if \mathbb {C} = \mathsf {Cat} this is precisely the ordinary comma category.

A 2-limit sketch is a small 2-category \mathcal {T} equipped with a (small) set of cylinders \Theta . A model of the sketch in \mathbb {C} is a 2-functor \mathcal {T} \to \mathbb {C} which carries each cylinder in \Theta to a limit cylinder. We write the category of models and natural transformations \mathsf {Mod}(\mathcal {T},\mathbb {C}) (leaving the set of cylinders implicit). If \mathbb {C} = \mathsf {Cat}, we write simply \mathsf {Mod}(\mathcal {T}).

Let \mathcal {T} be a 2-limit sketch. Then the functor \mathsf {Mod}(\mathcal {T},\mathbb {C}) \to [\mathbb {C}^\mathrm {op},\mathsf {Mod}(\mathcal {T})] given by A \in \mathsf {Mod}(\mathcal {T},\mathbb {C}) \mapsto (C \mapsto (S \in \mathcal {T} \mapsto \mathbb {C}(C,A(S)))) is fully faithful, and its essential image consists of those functors F so that each F(-)(S) : \mathbb {C}^\mathrm {op} \to \mathsf {Cat}, S \in \mathcal {T} is representable,

Let \mathcal {T},\mathcal {T}' be limit sketches and suppose given a functor F: \mathsf {Mod}(\mathcal {T}) \to \mathsf {Mod}(\mathcal {T}') which preserves limits and is accessible, that is it preserved \kappa -filtered colimits for some \kappa . Then F admits a left adjoint L.

In particular, F(A)(S) = \mathsf {Mod}(\mathcal {T})(L(y(S)),A)

Let \mathcal {C} be a 2-category with pullbacks. Then an internal pseudocategory in \mathcal {C} (or simply a pseudocategory in \mathcal {C}) consists of the following data:

1. Two objects C_0, C_1 \in \mathcal {C}
2. Morphisms d,c: C_1 \to C_0, e: C_0 \to C_1 so that de = ce = 1_{A_0}
.
3. A morphism m: C_1 \times _{C_0} C_1 \to C_1, so that dm = d\pi _2, cm = c\pi _1
4. Isomorphism 2-cells \alpha : m(1_{C_1} \times _{C_0} m) \to m(m \times _{C_0} 1_{C_1}), \lambda : m \langle ec, 1_{C_1} \rangle \to 1_CA_1
, and \rho : m\langle 1_{A_1}, ed \rangle \to 1_{A_1}
5. Satisfying the following equations:
   1. d \circ \lambda = 1_d = d \circ \rho
   2. c \circ \lambda = 1_c = c \circ \rho
   3. d \circ \alpha = 1_{d\pi _3}, c \circ \alpha = 1_{c\pi _1}
   4. \lambda \circ e = \rho \circ e
6. And so that the following diagrams commute:

A homomorphism or strict functor of pseudocategories A \to B is a pair of morphisms F_1: A_1 \to B_1, F_0: A_0 \to B_0 which commute with all the structure---that is, dF_1 = F_0d, F_1 \alpha = \alpha (F_1 \times _{F_0} F_1 \times _{F_0} F_1), and so on. There is a clear notion of natural transformation of homomorphisms. We write \mathsf {PsCat}_s(\mathbb {C}) for the 2-category of pseudocategories, homomorphisms and natural transformations in \mathbb {C}.

1. An internal pseudocategory in \mathsf {Cat} is, as mentioned above, a pseudo double category.
2. An internal pseudocategory A with A_0 terminal is the same thing as an internal pseudomonoid. If we fix a specific terminal object * and require A_0 = *, this is an isomorphism of 2-categories (both for the categories of pseudomorphisms and homomorphisms).
3. An internal pseudocategory with \alpha ,\lambda ,\rho identities is the same thing as a (strict) internal category. In particular, internal pseudocategories in discrete 2-categories are merely internal categories.

Let \mathsf {MonCat}_s be the 2-category of monoidal categories, strict monoidal functors and monoidal natural transformations. Note that this has pullbacks. Then the objects of \mathsf {PsCat}(\mathsf {MonCat}) are a stricter version of monoidal pseudo double categories: they are pseudo double categories C_1,C_0 where both C_1,C_0 carry a monoidal structure so that d,c,e,m are strict monoidal functors and \rho ,\lambda ,\alpha are monoidal natural transformations.

Let C = (C_0,C_1, d,c,e,m,\alpha ,\lambda ,\rho ) D = (D_0,D_1, d',c',e',m',\alpha ',\lambda ',\rho ') be internal pseudocategories in a 2-category \mathbb {C}. A pseudomorphism F: C \to D consists of the following data:

1. Morphisms F_0: C_0 \to D_0, F_1:C_1 \to D_1 in \mathbb {C}
2. Isomorphism 2-cells \mu : F_1m \to m(F_1 \times _{F_0} F_1) and \epsilon : F_1e \to eF_0

1. d'F_1 = F_0d, c'F_1 = F_0c
2. d' \circ \mu = 1_{F_0}d\pi _2, c' \circ \mu = 1_{F_0c\pi _1}
3. d' \circ \epsilon = 1_{F_0}, c'\circ \epsilon = 1_{F_0}

There exists a \mathsf {Cat}-limit sketch (\mathcal {T}_\mathsf {PsCat}, \Theta _\mathsf {PsCat}) whose strict models are internal pseudocategories, and whose strict natural transformations are homomorphisms.

Let \mathcal {M} be a monoidal category. A \mathcal {M}-actegory (also \mathcal {M}-module, \mathcal {M}-action) is a category \mathcal {C} equipped with a functor \cdot : \mathcal {M} \times \mathcal {C} \to \mathcal {C} called the action (written infix, M \cdot C), and natural isomorphisms \mu : (M \otimes M') \cdot C \to M \cdot (M' \cdot C), \eta : I \cdot C \to C satisfying the following coherence equations:

Let \mathbb {C} be a 2-category. A pseudomonoid action internal to \mathbb {C} consists of the following data: An internal pseudomonoid M, an object C, a morphism \cdot : M \times C \to C, natural isomorphisms \mu : \cdot (\otimes \times 1_C) \to \cdot (1_M \times \cdot ) and \eta : \cdot \langle e, 1_C \rangle \to 1_C, satisfying the coherence equations from Definition [efr-OFNV].

A strict homomorphism of actions is a pair F_m: M \to M', F_c: C \to C' so that F_m is a strictly monoidal functor and F_c preserves the action strictly, i.e F_m(M) \cdot F_c(C) = F_c(M \cdot C), F_c(\eta ) = \eta ', etc. Note that this makes sense even if the monoidal category or action is not itself strict.

There is a limit sketch \mathcal {T}_\mathsf {Act}, \Theta _\mathsf {Act} whose models are tuples (M,C,\cdot ) of a pseudomonoid M = (M,\otimes ,I), an object C and an action \cdot of M on C. The strict natural transformations between models are strictly linear functors.

If \mathbb {C} = \mathcal {C} 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.

In Reference [actegories-amthematician-capucci-gavranovic], 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 \mathsf {MonCat} agree with their notion: such an action consists of a braided monoidal category \mathcal {M}, an ordinary monoidal category \mathcal {C}, a monoidal functor \mathcal {M} \times \mathcal {C} \to \mathcal {C}, and monoidal natural transformations \mu ,\eta satisfying the coherence equations for an actegory.

(Note that since the functor \otimes appears on one side of \mu , to speak of \mu being a monoidal natural transformation we must have a monoidal structure on \otimes ---this makes \mathcal {M} into a braided monoidal category).

For the case of symmetric monoidal actegories, since \mathsf {SymMonCat} is cocartesian, \mathsf {PsMon}(\mathsf {SymMonCat}) \simeq \mathsf {SymMonCat}. Given two symmetric monoidal categories \mathcal {M},\mathcal {C}, Reference [actegories-amthematician-capucci-gavranovic] show that an action is simply given by a strong symmetric monoidal functor f: \mathcal {M} \to \mathcal {C} (and in this case the action is M \cdot C = F(M) \otimes C). Unwinding this construction we see that \mathsf {Act}_s(\mathsf {SymMonCat}) is bi-equivalent to a category which has as objects strong (but not strict) symmetric monoidal functors C \to C', and morphisms squares

Let \mathbb {C} be a 2-category, and let (M,C,\cdot ), (N,D,\star ) be internal pseudomonoid actions. A pseudolinear map (or just linear) is a pair of maps F_m: M \to N, F_c: C \to D, equipped with a pseudomonoid pseudohomomorphism structure \phi , \tau on F_m and a natural isomorphism l: F_m(-) \cdot F_c(=) \to F_c(- \cdot =) satisfying the following coherence conditions (note that writing functors using element-notation like this is well-defined)

Let \mathbb {C} be a 2-category which admits pullbacks, products and comma objects. Then there is a functor \mathsf {\mathbb Para}: \mathsf {Act}(\mathbb {C}) \to \mathsf {PsCat}(\mathbb {C}) from the 2-category of pseudomonoid actions and pseudolinear maps to the 2-category of pseudocategories and pseudofunctors, with \mathsf {\mathbb Para}_\mathcal {M}(\mathcal {C})_0 = \mathcal {C} and \mathsf {\mathbb Para}_\mathcal {M}(\mathcal {C})_1 = \mathcal {M} \times \mathcal {C} \downarrow \mathcal {C}. This functor preserves strict maps, limits, and filtered colimits.

Let \mathbb {C} be a 2-category and let C,D : \mathcal {T}_\mathsf {PsCat} \to \mathbb {C} be pseudocategories, represented as models of the theory. Then a pseudonatural transformation F: C \to D which is strict on c,d,e and the limit projections \pi ^i_j is equivalently a pseudomorphism between the pseudocategories.

Now let (M,C), (N,D) : \mathcal {T}_\mathsf {Act} \to \mathbb {C} be pseudomonoid actions. Then a pseudonatural transformation F_m, F_c which preserves the product projections strictly is equivalently a pair of a pseudomorphism F_m : M \to N and a functor F_c: C \to D which is pseudolinear with respect to the action F_m(-) \cdot = of M on D.

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 [power-enriched-lawvere]. 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 \mathsf {Cat}-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 [varkor-bourke-ko-sketches] (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.

First recall that \mathsf {Mod}(\mathcal {T},\mathbb {C}) can be identified with the subcategory of [\mathbb {C}^\mathrm {op}, \mathsf {Mod}(\mathcal {T})] spanned by the levelwise representable presheaves. Note that for \mathcal {T} = \mathcal {T}_\mathsf {PsCat}, it suffices to verify representability for C_0,C_1 \in \mathcal {T}_\mathsf {PsCat}, since the rest are pullbacks of these, and pullbacks of representable presheaves are again representable since \mathbb {C} admits pullbacks.

Postcomposition with the previously-constructed functor \mathsf {Mod}(\mathcal {T}_\mathsf {Act}) = \mathsf {Act}(\mathsf {Cat}) \to \mathsf {PsCat}(\mathsf {Act}) = \mathsf {Mod}(\mathcal {T}_\mathsf {PsCat}) gives a functor [\mathbb {C}^\mathrm {op},\mathsf {Mod}(\mathcal {T}_\mathsf {Act})] \to [\mathbb {C}^\mathrm {op},\mathsf {Mod}(\mathcal {T}_\mathsf {PsCat})]. Clearly this functor preserves representability (again, since limits of representable functors are representable). This gives the desired functor \mathsf {Mod}_s(\mathcal {T}_\mathsf {Act}, \mathbb {C}) \to \mathsf {Mod}_s(\mathcal {T}_\mathsf {PsCat}, \mathbb {C}).

Under the identification of \mathsf {Mod}(\mathcal {T},\mathbb {C}) with a subcategory of [\mathbb {C}^\mathrm {op} \times \mathcal {T}, \mathsf {Cat}], it is clear that the pseudonatural transformations of models correspond to those pseudonatural transformations which are strict on morphisms in \mathbb {C}^\mathrm {op}. This implies the full category \mathsf {Mod}_p(\mathbb {C}) of models and pseudonatural transformations can be identified with the subcategory of [\mathbb {C}^\mathrm {op},\mathsf {Mod}_p(\mathcal {T},\mathsf {Cat})] spanned by those 2-functors which come from models---that is, A: \mathbb {C}^\mathrm {op} \to \mathsf {Mod}_p(\mathcal {T}) = \mathsf {Mod}_p(\mathcal {T},\mathsf {Cat}) must factor over \mathsf {Mod}(\mathcal {T}) \hookrightarrow \mathsf {Mod}_p(\mathcal {T}).

(To be clear: the morphisms in [\mathbb {C}^\mathrm {op}, \mathsf {Mod}_p(\mathcal {T})] are strictly natural transformations between functors \mathbb {C}^\mathrm {op} \to \mathsf {Mod}_p(\mathcal {T}), but each component A(C) \to B(C) 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 [\mathbb {C}^\mathrm {op}, \mathsf {Mod}_p(\mathcal {T}_\mathsf {PsCat})] which are valued in pseudofunctors, and analogously for pseudolinear morphisms and maps [\mathbb {C}^\mathrm {op}, \mathsf {Mod}_p(\mathcal {T}_\mathsf {Act})]. Hence it suffices to observe that the \mathsf {Cat}-valued version \mathsf {Mod}(\mathcal {T}_\mathsf {Act}) \to \mathsf {Mod}(\mathcal {T}_\mathsf {PsCat}) carries pseudolinear maps to pseudofunctors.

Note that \mathsf {\mathbb Para}: \mathsf {Mod}(\mathcal {T}_\mathsf {Act}) \to \mathsf {Mod}(\mathcal {T}_\mathsf {PsCat}) 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 [\mathcal {T}_\mathsf {PsCat}, \mathsf {Cat}]. Hence \mathsf {\mathbb Para} commutes with filtered colimits, and is in particular accessible. Hence it admits a left adjoint L (M,C) \mapsto \operatorname {\mathrm {Hom}}_{\mathsf {Mod}(\mathcal {T}_\mathsf {Act})(L(-), (M,C))} where L: \mathcal {T}_\mathsf {PsCat}^\mathrm {op} \to [\mathcal {T}_\mathsf {Act},\mathsf {Cat}] is the restriction of the left adjoint of \mathsf {\mathbb Para}. Both forming the hom-category \operatorname {\mathrm {Hom}}(-, (M,C)) and precomposing with L are 2-functors, and as such preserve pseudonatural transformations. Thus it only remains to observe that \mathsf {\mathbb Para} also preserves the partial strictness property, but this clear.

If \mathbb {C} \to \mathbb {D} 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).

Let \mathcal {C} be a monoidal category. A selection relation on an object A \in \mathcal {C} is a relation \epsilon \subseteq \mathcal {C}(I,A) \times \mathcal {C}(A,I). We write \epsilon (a,k) for the statement (a,k) \in \epsilon .

Selection relations are ordered by inclusion, and so form a (posetal) category, which we denote \mathbb {S}_\mathcal {C}(A). Given f: A \to B, we define the pushforward on selection relations by f_*\epsilon = \{(fx,k) \mid (x, kf) \in \epsilon \}. In other words, f_*\epsilon (y,k) if and only if there exists x: I \to X so that fx = y and \epsilon (x, kf).

It is clear that pushforward is monotone, so that this defines a functor \mathbb {S}_\mathcal {C} : \mathcal {C} \to \mathsf {Cat}

Of course, there is an equally good "pull-back" operation on selection relations given by \{(x,kf) \mid (fx, k) \in \epsilon \} (this is the pushforward in \mathbb {S}_{\mathcal {C}^\mathrm {op}}). 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 \mathcal {C} \to \mathsf {\mathbb Cat}, the category of categories and profunctors, where we say two selection relations \epsilon \in \mathbb {S}_X, \epsilon ' \in \mathbb {S}_Y are related by f :X \to Y if \epsilon (x, kf) \Rightarrow \epsilon '(fx, k).

However, we will stick with the pushforward definition for now, since it is conceptually simpler and good enough for our purposes.

1. Let \mathcal {C} = \mathsf {Optic}(\mathsf {Set}). There is a selection function \mathrm {argmax}_X \in \mathbb {S}({\mathbb {R} \choose X}) defined by \mathrm {argmax}(x,f) if and only if x is a maximum of f. (Identifying maps I \to {\mathbb {R} \choose X} with points x \in X, and maps {\mathbb {R} \choose X} \to I with functions X \to \mathbb {R})
2. In the same category, for each r \in \mathbb {R}, there is a selection function given by \epsilon (x,f) \Leftrightarrow f(x) \geq r. This corresponds to satisficing at the value r (Reference [simon-environment-satisficing])---that is, selecting any strategy which achieves this value or greater.
3. For \mathcal {C} = \mathsf {Optic}(\mathcal {C}'), with \mathcal {C}' semiCartesian, there is a selection function \epsilon (x,k) \Leftrightarrow kx = x (using the same identification as above). The agents with this selection function are called predicting agents in Reference [bolt-hedges-winschel-predicting]. These agents attempt to predict the value the environment will return to them.
4. For \mathcal {C} = \mathsf {Bun}^\mathrm {fop}, 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 TX given by \epsilon (x,k) if and only if k(x) = 0---that is, if x is a fixpoint of the dynamical system identified by k: X \to TX.
5. Let \mathcal {A} \to \mathcal {C} be a dynamical systems theory and work in the category of lenses. Suppose the monoidal structure on \mathcal {A} is Cartesian, so that a lens I \to {A \choose X} is the same as a map * \to X. Note that I = T(*) has a distinguished section given by the identity. Then there is a selection function on each object TX where a lens I \to TX, given by x: * \to X, is in equilibrium with respect to a lens TX \leftrightarrows I if x is a trajectory between those systems---that is, if it is an equilibrium state of the smooth dynamical system TX \leftrightarrows I. This subsumes the two previous examples.

Let \epsilon \in \mathbb {S}(X), \epsilon ' \in \mathbb {S}(Y) be selection relations. Their Nash product \epsilon \boxtimes \epsilon ' \in \mathbb {S}(X \otimes Y) is given by \{(x \otimes y, k) \mid \epsilon (x,k(1_X \otimes y)), \epsilon '(y, k(x \otimes 1_Y))\}

Consider the selection relation \mathrm {argmax}_X \boxtimes \mathrm {argmax}_Y on {\mathbb {R} \times \mathbb {R} \choose X \times Y}. A point (x,y) is an equilibrium for a function k: X \times Y \to \mathbb {R}^2 if and only if x maximizes k_1(-,y) and y maximizes k_2(x,-). In other words, if x,y is a Nash equilibrium (Reference [nash-noncoop-games]) in the usual sense for the game with payoff matrix k.

The Nash product \mathbb {S}(X) \times \mathbb {S}(Y) \to \mathbb {S}(X \otimes Y), along with the map * \to \mathbb {S}(I) given by the full set S(I) \times S(I) equips \mathbb {S} with a lax monoidal structure.

Since \mathbb {S} is lax monoidal, its Grothendieck construction \int \mathbb {S}_\mathcal {C} acquires a monoidal structure (Reference [moeller-vasilakopoulou]). We write this category \mathcal {C}_\mathbb {S}---explicitly, it is given as follows:

1. The objects are pairs (X,\epsilon ) where X \in \mathcal {C} and \epsilon \in \mathbb {S}(X) is a selection relation on it
2. The morphisms are morphisms f: X \to Y so that, for every x: I \to X, k: Y \to I, \epsilon (x, kf) \Rightarrow \epsilon (fx, k)
3. The monoidal structure is given by (X,\epsilon ) \otimes (Y,\epsilon ') = (X \otimes Y, \epsilon \boxtimes \epsilon ')

Let \mathcal {C} be a semiCartesian symmetric monoidal category. The symmetric monoidal double category of open games in \mathcal {C} is \widetilde {\mathsf {Game}}(\mathcal {C}) = \mathsf {\mathbb Para}_{\mathsf {Optic}(\mathcal {C})_\mathbb {S}}(\mathsf {Optic}(\mathcal {C})).

Let \mathcal {D} be a symmetric monoidal stochastic module fibration over the Markov category \mathcal {C}. Recall that \mathsf {SLens}(\mathcal {D}) acquires a symmetric monoidal structure. Denote as usual \mathsf {SLens}(\mathcal {D})|_\mathrm {det} = \mathsf {SLens}(\mathcal {D}) \times _{\mathcal {C}} \mathcal {C}_\mathrm {det}. Note that this is stable under the monoidal product, and acts on \mathsf {SLens}(\mathcal {D}) via the inclusion.

Then the category of open games in \mathcal {D} is the category \widetilde {\mathsf {Game}} = \mathsf {Para}_{(\mathsf {SLens}(\mathcal {D})|_\mathrm {det})_\mathbb {S}}(\mathsf {SLens}(\mathcal {D})).

When \mathcal {D} \to \mathcal {C} is a Markov prefibration, we overload the notation by writing \widetilde {\mathsf {Game}}(\mathcal {D}) = \mathsf {Game}(\mathcal {D}|_\mathrm {det}).

Let X,Y be objects of a symmetric monoidal category \mathcal {C}. A context for X,Y is a tuple M \in \mathcal {C}, s: I \to X \otimes M, k: Y \otimes M \to I. We denote the set of contexts \mathrm {Ctx}(X,Y).

Given a morphism f: P \otimes X \to Y, and a context c = (M,s,k), the costate k(f \otimes 1_M)(1_{P} \otimes s) will be called the induced costate.