Reference Actegories for the Working Amthematician [actegories-amthematician-capucci-gavranovic]

Abstract

  • December 11, 2023
  • Matteo Capucci, Bruno Gavranović

Actions of monoidal categories on categories, also known as actegories, have been familiar to category theorists for a long time, and yet a comprehensive overview of this topic seems to be missing from the literature. Recently, actegories have been increasingly employed in applied category theory, thereby encouraging an effort to fill this gap according to the new needs of these applications. This work started as an investigation of the notion of monoidal actegory, a compatible pair of monoidal and actegorical structures, and ended up including a sizable reference on the elementary theory of actegories. We cover basic definitions and results on actegories and biactegories, spelling out explicitly many folkloric definitions, including their tensor product and their hom-tensor adjunction. We give new definitions of actegories with monoidal, braided monoidal and symmetric monoidal structure. In the last section, we provide three Cayley-like classification results for these structures. 

 @article
{actegories-amthematician-capucci-gavranovic, title={Actegories for the Working Amthematician}, url={http://arxiv.org/abs/2203.16351}, DOI={10.48550/arXiv.2203.16351}, number={arXiv:2203.16351}, publisher={arXiv}, author={Capucci, Matteo and Gavranović, Bruno}, year={2023}, month={Dec} }

Backlinks

Actegories [efr-NI76]

Definition Actegory [efr-OFNV]

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:

Actegories are a very natural concept, and as one would expect their history goes back a long way. The concept has been considered by Benabou all the way back in Reference [benabou-bicategories], and used many times since them. We will not delve too deeply into their theory here---see Reference [actegories-amthematician-capucci-gavranovic] for a thorough treatment.

Just as pseudo double categories, actegories really make sense in every 2-category, as a weakening of monoid actions.

Definition Pseudomonoid action [efr-IOLJ]

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.

Proposition [efr-6S13]

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.

Proof

As above, there is nothing difficult about this. Let \mathcal {T}_\mathsf {Act} contain as a subcategory the sketch of pseudomonoids \mathcal {T}_\mathsf {PsMon}, along with three additional objects C, MC, MMC, projections \pi ^1_M, \pi ^1_C: MC \to M,C, \pi ^2_{M_1}, \pi ^2_{M_2}, \pi ^2_{C} : MMC \to M, M, C, and limit cones expressing these as the products M \times C and M \times M \times C respectively, a morphism \cdot : MC \to C, and isomorphism 2-cells \eta : 1_C \Rightarrow \cdot \langle I, 1_C \rangle and \mu : \cdot \langle \pi ^2_{M_1}, \cdot \langle \pi ^2_{M_2}, \pi ^2_C \rangle \rangle \to \cdot \langle \otimes \langle \pi ^2_{M_1}, \pi ^2_{M_2} \rangle , \pi ^2_C \rangle , subject to the coherence equations in Definition [efr-OFNV].

Example Discrete pseudomonoid [efr-I33G]

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.

Example Monoidal and symmetric monoidal actegories [efr-ARIX]

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

of symmetric monoidal functors, where the top map is strict, and which commute strictly.

Again, as expected, the notion of strict linear morphism is usually too strict. One can expect an equation like F_m(M) \cdot F_c(C) = F_c(M \cdot C) to hold only up to coherent isomorphism. Let us now make this clear:

Definition Pseudolinear map [efr-L5QG]

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)

In Reference [actegories-amthematician-capucci-gavranovic], these are defined in two steps: first linear functors for the same \mathcal {M} (those with F_m = 1_\mathcal {M}) are considered. Then, given a strong monoidal \mathcal {M} \to \mathcal {N}, a functorial assignment of an \mathcal {M}-action to every \mathcal {N}-action is constructed, and the full category of actions is defined as the Grothendieck construction of this. There is nothing preventing this from working in the setting of a general 2-category, and it is straightforward to verify that our notion of linear morphism of actions agrees with theirs.

Example Actegories as pseudomonoid actions [efr-9LCF]

As discussed above, \mathsf {Act}(\mathsf {Cat}) is the locally full subcategory of the 2-category of actegories and pairs of strong monoidal and lax linear functors constructed in Reference [actegories-amthematician-capucci-gavranovic] (see Proposition 3.6.5) spanned by the pseudolinear (or "strongly linear") functors.

Example Monoidal and symmetric monoidal actegories [efr-ARIX]

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

of symmetric monoidal functors, where the top map is strict, and which commute strictly.

Proposition [efr-A6YR]

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.

Proof

We have essentially already seen that the deterministic part \mathsf {Optic}_\mathcal {C}(\mathcal {D})|_\mathrm {det} \to \mathcal {C}_\mathrm {det} is a fibration, with maps {A \choose X} \leftrightarrows {B \choose X} over X given by X \cdot B \to A, and with the pullback functors acting by reparametrization. Since a map X \to M \otimes Y with deterministic marginal on Y is always equal to the independent pairing of X \to M, X \to Y, we can slide the former through and identify each optic over a given X \to Y with a map X \times B \to A, and note that this map is conversely an invariant of an optic, since it is obtained by postcomposing with {B \choose Y} \to {B \choose *}.

Given M \to X, objects {A \choose X}, {B \choose X}, and a map over M classified by M \cdot B \to A, a section s :X \to M act by reparametrization. It is clear that this gives the structure of a stochastic module.

The functor from optics takes f: X \to M \otimes Y, g: M \cdot B \to A to the span X \leftarrow M \otimes X \otimes Y \to Y equipped with the obvious map (M \otimes X \otimes Y) \cdot B \to A that simply forgets X,Y. Note that every chart is equivalent to one of this form (given X \leftarrow M' \to Y and \phi : M' \cdot B \to A, the map M' \to M' \otimes X \otimes Y exhibits the required equivalence), hence the functor is full.

Moreover, note that each chart is associated with a well-defined optic, given by the maps X \to M \to M \otimes Y, M \cdot B \to A. Is is straightforward to see both of these maps are preserved by chart equivalence. This gives an inverse to the functor, proving it is faithful. Since it is identity on objects, this finishes the argument.

In the symmetric monoidal case, it is immediately clear that the fibration on \mathcal {C}_\mathrm {det} is symmetric monoidal. Since the action is symmetric monoidal, given maps X \to M, X \to M' and M \cdot B \to A, M' \cdot B' \to A', it is clear that composing to get maps X \cdot B \to A, X \cdot B' \to A', then tensoring and composing with the diagonal to get X \cdot (B \otimes B' ) \to A \otimes A', gives the same map as tensoring, then using the map X \to M \otimes M'. Hence we have a symmetric monoidal module. It's straightforward to see the functor is symmetric monoidal, and that finishes the argument.

Proposition [efr-002N]

There is an equivalence of bicategories between

  1. (+,-)-fibrations \mathcal {C} \to B\mathcal {M} which are locally discrete fibrations.
  2. Pseudofunctors B\mathcal {M} \to \mathsf {Cat}
  3. Actions of \mathcal {M}

Proof

The latter two are equivalent almost by definition, see eg Reference [actegories-amthematician-capucci-gavranovic].

By Definition [efr-002U], clearly pseudofunctors B\mathcal {M} \to \mathsf {Cat} are equivalent to some subcategory of the (+.-)-fibrations. But a fibration classifies a pseudofunctor into 1-categories if and only if its fibers are equivalent to 1-categories, if and only if the hom-categories of the fibers are equivalent to discrete categories. Since the hom-categories of the fibers are the fibers of \mathcal {C}(X,Y) \to B\mathcal {M}(*,*), this is true if and only if these fibrations are discrete, concluding the proof.

The Para Construction as a double category [efr-002H]

In many different situations, we want to understand some morphism as parametrized by some data. For example, in machine learning one tries to find a function f: X \to Y with some desirable behaviour by choosing a parametrization F: P \times X \to Y and searching for some p \in P so that F(p,-) has this behaviour (for example by gradient descent on p).

In situations where we want to understand F as being built up as a composite of multiple functions (for example, the layers of a neural network), it is convenient to introduce a category of parametrized morphisms, where the composition combines the parameter spaces of each composite map. We can do this in a general setting with the following definition:

Definition [efr-002G]

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.

The fact that the parametrization object may live in a different category than the domain and codomain objects, which initially seems like a superfluous generalization, is in fact highly useful. For example, we will often want to consider parametrized morphisms where the parameter space is decorated with some additional data, for example a probability distribution. In most such cases, the category of spaces decorated with such data will act on the category of spaces without such data (simply by forgetting the data and tensoring), and hence we can realize such parametrized morphisms as examples of this \mathsf {Para} construction. We may also want only Euclidean parameter space (so that we can run gradient descent simply), but allow the parametrized morphism to go between more general manifolds.

\mathsf {Para}, and generalizations of it, have been introduced many times. One of the earliest occurrences is by Hermida and Tennent, Reference [hermida-monoidal-indeterminates], in the special case of a symmetric monoidal category acting on another such via a functor i: \mathcal {C} \to \mathcal {D} (by a result of Capucci and Gavranović, Reference [actegories-amthematician-capucci-gavranovic], every "structure-preserving" action of a symmetric monoidal category on another has this form). Hermida and Tennent actually give a very intuitive universal property of \mathsf {Para}_\mathcal {C}(\mathcal {D}): it is freely generated by adding a morphism I \to i(C) for each C \in \mathcal {C}, subject to the equations that this must be a monoidal natural transformation. This type of idea actually goes all the way back to Pavlović in Reference [pavlovic-categorical-names].

The notation \mathsf {Para} was introduced inReference [backprop-as-functor], where the special case of parametrized morphisms of euclidean spaces was used to study gradient descent. In Reference [towards-cybercat], a bicategorical variant (replacing the quotient by isomorphism in the above variant in the obvious way) is introduced.

Example [efr-MN7C]

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.

Example [efr-WEST]

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).

Example [efr-QH7O]

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.

[efr-KL7V]

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})

There is a very natural double categorical version of \mathsf {Para}, where the vertical morphisms are just the unparametrized morphisms of \mathcal {C}, and the 2-cells with this boundary:

are morphisms M \to N (if f is parametrized by M, g by N) so that the square
commutes.

This notion is not original---we learned of it from Matteo Capucci---but it seems to have remained somewhat on the level of folklore. Our main goal for this section will be to construct this (pseudo) double category, and show that it is functorial---that is, given a homomorphism of actions, there is an induced functor between double categories. In fact, we will show that this construction works in any 2-category. Later we will apply it to a pseudomonoid action in (strict) double categories to construct a pseudocategory internal to double categories---our triple category of bimachines.

Theorem [efr-55CC]

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

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.

Remark [efr-HUEE]

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.

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

Myers' theory of categorical systems theory (see § [efr-0023]) gives a rich categorical structure to a wide variety of types of dynamical system. The central idea can be summarized by saying that there are two different, but tightly related, notions of morphism at play in dynamical systems. Open dynamical systems themselves involve a bidirectional information flow, captured by the notion of lens {X' \choose X} \leftrightarrows {Y' \choose Y}, and composition of such lenses describes the composition of subsystems into systems. But morphisms between systems are unidirectional, captured by the notion of chart {X' \choose X} \rightrightarrows {Y' \choose Y},. Algebraically, the relationship between these two notions is that they assemble into a double category, which indexes the category of systems.

There is another double category involving lenses which has been considered in the categorical study of systems. That is the double category of parametrized morphisms, applied to the monoidal category of lenses. These have been studied as an abstraction for gradient descent in several papers, see eg Reference [bruno-etal-categorical-learning-2021], Reference [bruno-thesis-fundamental-components], (the idea goes back to Reference [backprop-as-functor]). Given any action of a monoidal category \mathcal {M} on another category \mathcal {C} (most simply, if \mathcal {C} is monoidal it acts on itself,) we obtain a category of parametrized morphisms f: X \cdot P \to Y (where P \in \mathcal {M}, X, Y \in \mathcal {C},) and these turn out to be extremely useful. We will mention two applications:

  1. A parametrized lens {P \choose P} \otimes {X \choose X} \to {Y \choose Y} is essentially what is called a learner in Reference [backprop-as-functor]---it contains the information necessary to compute a new parameter value p' given an existing p \in P and a sample pair x \in X, y \in Y, as well as the additional information required to compose such things. Thus the functoriality of backpropagation can be derived from two facts: the reverse derivative defines a monoidal functor \mathsf {Euc} \to \mathsf {Lens}(\mathsf {Euc}), and the construction \mathsf {Para}(-), taking a category to its category of parametrized maps, is itself functorial. This viewpoint has been significantly developed in Reference [bruno-etal-categorical-learning-2021] and other papers.
  2. An open game, in the sense of Hedges Reference [hedges-etal-comp-gametheory], is almost the same thing as a parametrized lens {\Sigma ' \choose \Sigma } \otimes {S \choose X} \leftrightarrows {R \choose Y}, equipped with a subset E \subset \Sigma \times \mathsf {Set}(\Sigma ,\Sigma '). Given a context for the game---that is, a state x \in X (describing the state of information when the decision is made) and a continuation Y \to R (describing how decisions y \in Y map to outcomes r \in R,) we obtain a function k: \Sigma \to \Sigma ', and we say \sigma \in \Sigma is an equilibrium strategy if (\sigma ,k) \in E. Since \mathsf {Set}(\Sigma ,\Sigma ') = \mathsf {Lens}(\mathsf {Set})({\Sigma ' \choose \Sigma },I), and \Sigma = \mathsf {Lens}(\mathsf {Set})(I,{\Sigma ' \choose \Sigma }), this neatly captures the extra data of an open game in terms of the category of lenses. The potential of this idea as a generalized approach to "cybernetic systems" is explored in Reference [towards-cybercat].

In this chapter, we will develop the theory of the category \mathsf {Para} of parametrized maps. We will begin by reviewing the existing literature briefly. We will describe a double categorical version of this category---this does not seem to have appeared in the literature yet, although it has been folklore for at least a few years (and there is nothing complicated about this construction, certainly). The remainder of this chapter will be dedicated to lifting this construction to a generic 2-category \mathbb {C} (with the above being the specialization to \mathbb {C} = \mathsf {Cat} ). We will derive this lifting using the machinery of 2-category theory. We will see how this generalization accounts for much structure which can be seen to exist on \mathsf {Para}, such as its symmetric monoidal structure (assuming \mathcal {M},\mathcal {C} are symmetric monoidal). But the true application of this will be in § [efr-ZRUZ], where we use this to construct a triple category of open dynamical systems.

The definition of the double category \mathsf {\mathbb Para} involves two types of categorical structure with which the reader may be unfamiliar---actegories, which are the input to the construction, and pseudo double categories, which are the output. Since we will shortly introduce the abstract internal versions of these, internal pseudomonoid actions and internal pseudocategories, we will not give a separate introduction here. The reader who is unfamiliar with these should refer to Reference [actegories-amthematician-capucci-gavranovic] for actegories, and Reference [shulman-monfibs] for (pseudo) double categories. A reader who simply needs a definition may look at Definition [efr-OFNV] and Example [efr-ZRUY].