nLab
action of a category on a set

Definition

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

Action as a functor

An action of a category $C$ on a set $S$ is nothing but a functor $\rho :C\to$ Set.

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

$$S={\bigsqcup }_{c\in \mathrm{Obj}(C)}\rho (c).$$S = \sqcup_{c \in Obj(C)} \rho(c) \,.

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

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

and the the action of $f$ on $s\in \rho (c)$ under $\rho$ is

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

In the case that $C$ has just a single object $•$ the category $C$ is just a monoid (might for instance be a group), there is just a single set $S=\rho (•)$ 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 $\rho :G\to \mathrm{Aut}(S)$ is a group action on a set $S$, 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 $S={\bigsqcup }_{c\in \mathrm{Obj}(c)}\rho (c)$ of the set $S$ into subsets corresponding to objects of the category $C$ can equivalently be encoded in a map of sets

$$\lambda :S\to \mathrm{Obj}(C)$$\lambda : S \to Obj(C)

which sends each element of $S$ to the object of $c$ it corresponds to under the action.

(In the case that our category $C$ 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 $C$ itself comes with maps to $\mathrm{Obj}(C)$: the source map $s$ and target map $t$, which are suggestively drawn as a span in Set by writing:

$$\begin{array}{ccc}& & \mathrm{Mor}(C)\\ & {}^s\swarrow & & {\searrow }^t\\ \mathrm{Obj}(C)& & & & \mathrm{Obj}(C)\end{array}.$$\array{ && Mor(C) \\ & {}^{s}\swarrow && \searrow^{t} \\ Obj(C) &&&& Obj(C) } \,.

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

$$\begin{array}{ccc}& & S{}_{\lambda }{\times }_s\mathrm{Mor}(C)\\ & {}^{{\mathrm{pr}}_1}\swarrow & & {\searrow }^{{\mathrm{pr}}_2}\\ S& & & & \mathrm{Mor}(C)\\ & {\searrow }^{\lambda }& & {}^s\swarrow & & {\searrow }^t\\ & & \mathrm{Obj}(C)& & & & \mathrm{Obj}(C)\end{array}.$$\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 $C$ on $S$ sends every element in this fiber product, which is a pair

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

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

$$\begin{array}{ccc}& & S{}_{\lambda }{\times }_s\mathrm{Mor}(C)\\ & \swarrow & & {\searrow }^{t\circ {\mathrm{pr}}_2}\\ \mathrm{pt}& & {\downarrow }^{\rho }& & \mathrm{Obj}(C)\\ & \nwarrow & & {\nearrow }_{\lambda }\\ & & S\end{array}$$\array{ && S {}_\lambda \times_s Mor(C) \\ & \swarrow && \searrow^{t \circ pr_2} \\ pt &&\downarrow^{\rho}&& Obj(C) \\ & \nwarrow && \nearrow_{\lambda} \\ && S }

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

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

the set of composable morphisms $\mathrm{Mor}(C){}_t{\times }_s\mathrm{Mor}(C)$ is itself the tip of a span arising from composing the span of $C$ with itself by pullback:

$$\begin{array}{ccccc}& & & & \mathrm{Mor}(C){}_t{\times }_s\mathrm{Mor}(C)\\ & & & \swarrow & & \searrow \\ & & \mathrm{Mor}(C)& & & & \mathrm{Mor}(C)\\ & {}^s\swarrow & & {\searrow }^t& & {}^s\swarrow & & {\searrow }^t\\ \mathrm{Obj}(C)& & & & \mathrm{Obj}(C)& & & & \mathrm{Obj}(C)\end{array}$$\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 $C$ is a morphism from this composed span to the original span

$$\begin{array}{ccc}& & \mathrm{Mor}(C){}_t{\times }_s\mathrm{Mor}(C)\\ & {}^{s\circ {\mathrm{pr}}_1}\swarrow & & {\searrow }^{t\circ {\mathrm{pr}}_2}\\ \mathrm{Obj}(C)& & {\downarrow }^{\circ }& & \mathrm{Obj}(C)\\ & {}^s\nwarrow & & {\nearrow }_t\\ & & \mathrm{Mor}(C)\end{array}.$$\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 $S{}_{\lambda }{\times }_s\mathrm{Mor}(C){}_t{\times }_s\mathrm{Mor}(C)$ obtained by composing the anchor map span with two copies of the span of $C$ back to the anchor map span

$$\begin{array}{ccc}& & S{}_{\lambda }{\times }_s\mathrm{Mor}(C){}_t{\times }_s\mathrm{Mor}(C)\\ & {}^{}\swarrow & & {\searrow }^{t\circ {\mathrm{pr}}_3}\\ \mathrm{pr}& & \downarrow & & \mathrm{Obj}(C)\\ & {}^s\nwarrow & & {\nearrow }_{\lambda }\\ & & S\end{array}.$$\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 $\rho$, which is nothing but the functoriality of $\rho$ in the above description, says precisely that these two morphisms coincide.

Abstractly this says that

• a (small) category is a monad in spans in Set;

• the action of a category on a set is an algebra for this monad.

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