action of a category on a set



Given a (small) category CC and given a set SS there are (at least) the following two equivalent ways to define an action of CC on SS.

Action as a functor

An action of a category CC on a set SS is nothing but a functor ρ:C\rho : C \to Set.

The particular set SS that this functor defines an action on is the disjoint union of sets that the functor assigns to the objects of CC:

S= cObj(C)ρ(c). S = \sqcup_{c \in Obj(C)} \rho(c) \,.

Given an element sSs \in S which sits in the subset ρ(c)S\rho(c) \subset S associated with the object cc of CC, it is acted on by all morphisms cfdc \stackrel{f}{\to} d in CC whose source is cc. By the definition of functor every such morphism defines a map of sets

ρ(f):(ρ(c)S)(ρ(d)S) \rho(f) : (\rho(c) \subset S) \to (\rho(d) \subset S)

and the the action of ff on sρ(c)s \in \rho(c) under ρ\rho is

ρ(f):(sρ(c))(ρ(f)(s)ρ(d)). \rho(f) : (s \in \rho(c)) \mapsto (\rho(f)(s) \in \rho(d)) \,.

In the case that CC has just a single object \bullet the category CC is just a monoid (might for instance be a group), there is just a single set S=ρ()S = \rho(\bullet) and we recover the ordinary notion of a monoid or group acting on a set.

Indeed this generalizes the instance (the motivating example for the notion of action) where ρ:GAut(S)\rho:G\rightarrow \mathbf{Aut}(S) is a group action on a set SS, since the notion of coproduct is a generalization of the notion of automorphism group since naively a cardinal is an isomorphism class of sets and the notion of coproduct in turn generalizes that of cardinal ( see there).

Action as an algebra for a monad

An equivalent perspective on the above situation is often useful. To motivate this, notice that the decomposition S= cObj(c)ρ(c)S = \sqcup_{c \in Obj(c)} \rho(c) of the set SS into subsets corresponding to objects of the category CC can equivalently be encoded in a map of sets

λ:SObj(C) \lambda : S \to Obj(C)

which sends each element of SS to the object of cc it corresponds to under the action.

(In the case that our category CC is a groupoid or even a Lie groupoid this map may be familiar as the anchor map or moment map of the action.)

But also the category CC itself comes with maps to Obj(C)Obj(C): the source map ss and target map tt, which are suggestively drawn as a span in Set by writing:

Mor(C) s t Obj(C) Obj(C). \array{ && Mor(C) \\ & {}^{s}\swarrow && \searrow^{t} \\ Obj(C) &&&& Obj(C) } \,.

Recall from the above discussion that a morphism f:cdf : c \to d in CC could act on an element sSs \in S if the image of ss under the anchor map λ\lambda coincides with the source of ff, i.e. with the image of ff under the source map ss. Formally this means that the pairs of elements of SS and morphisms of CC which can be paired by the action live in the pullback set S λ× sMor(C)S {}_\lambda \times_s Mor(C) (the fiber product):

S λ× sMor(C) pr 1 pr 2 S Mor(C) λ s t Obj(C) Obj(C). \array{ && S {}_\lambda \times_s Mor(C) \\ & {}^{pr_1}\swarrow && \searrow^{pr_2} \\ S && && Mor(C) \\ & \searrow^{\lambda}& & {}^{s}\swarrow && \searrow^{t} \\ && Obj(C) &&&& Obj(C) } \,.

Above we have seen that the aciton of CC on SS sends every element in this fiber product, which is a pair

(sρ(c)S,(cfd)Mor(C)) (s \in \rho(c) \subset S, (c \stackrel{f}{\to} d) \in Mor(C))

to an element ρ(f)(s)ρ(d)\rho(f)(s) \in \rho(d). So this is a map of sets ρ:S λ× sCS\rho : S {}_\lambda \times_s C \to S. But a special such map, in that it satisfies a couple of conditions. One condition is that sρ(c)s \in \rho(c) is taken to ρ(d)\rho(d) by f:cdf : c \to d. This can be encoded by saying that ρ\rho extends to a morphism of spans from the pullback span above back to SS:

S λ× sMor(C) tpr 2 pt ρ Obj(C) λ S \array{ && S {}_\lambda \times_s Mor(C) \\ & \swarrow && \searrow^{t \circ pr_2} \\ pt &&\downarrow^{\rho}&& Obj(C) \\ & \nwarrow && \nearrow_{\lambda} \\ && S }

But ρ\rho satisfies yet another compatibility condition: so far we have only used the source-target mathcing condition of the functor ρ:CSet\rho : C \to Set. There is also its functoriality, i.e. its respect for composition.

But composition in the category CC is itself naturally expressed in terms of morphisms of spans:

the set of composable morphisms Mor(C) t× sMor(C)Mor(C) {}_t \times_s Mor(C) is itself the tip of a span arising from composing the span of CC with itself by pullback:

Mor(C) t× sMor(C) Mor(C) Mor(C) s t s t Obj(C) Obj(C) Obj(C) \array{ &&&& Mor(C) {}_t\times_s Mor(C) \\ &&& \swarrow && \searrow \\ && Mor(C) &&&& Mor(C) \\ & {}^s\swarrow && \searrow^t && {}^s\swarrow && \searrow^t \\ Obj(C) &&&& Obj(C) &&&& Obj(C) }

and the composition operation \circ in CC is a morphism from this composed span to the original span

Mor(C) t× sMor(C) spr 1 tpr 2 Obj(C) Obj(C) s t Mor(C). \array{ && Mor(C) {}_t \times_s Mor(C) \\ & {}^{s \circ pr_1}\swarrow && \searrow^{t \circ pr_2} \\ Obj(C) &&\downarrow^{\circ}&& Obj(C) \\ & {}^{s}\nwarrow && \nearrow_{t} \\ && Mor(C) } \,.

In total this gives us two different ways to map the total span with tip S λ× sMor(C) t× sMor(C)S {}_\lambda \times_s Mor(C) {}_t \times_s Mor(C) obtained by composing the anchor map span with two copies of the span of CC back to the anchor map span

S λ× sMor(C) t× sMor(C) tpr 3 pr Obj(C) s λ S. \array{ && S {}_\lambda \times_s Mor(C) {}_t \times_s Mor(C) \\ & {}^{}\swarrow && \searrow^{t \circ pr_3} \\ pr &&\downarrow && Obj(C) \\ & {}^{s}\nwarrow && \nearrow_{\lambda} \\ && S } \,.

The action property of ρ\rho, which is nothing but the functoriality of ρ\rho in the above description, says precisely that these two morphisms coincide.

Abstractly this says that

Generalizing this slightly, it should be possible to associate an action of a category CC on a category cC 0ρ(c)\coprod_{c\in C_0}\rho(c) to a functor ρ:CCat\rho:C\rightarrow \Cat with the expectation, that this then is just a module for CC as a monad.

Revised on June 30, 2017 10:28:06 by Anonymous (