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 $C$:

$$S = \bigsqcup_{c \in Obj(C)} \rho(c) \,.$$

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

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

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

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

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

$$\lambda : S \to Obj(C)$$

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 $Obj(C)$: the source map $s$ and target map $t$, which are suggestively drawn as a span in Set by writing:

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

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 $S {}_\lambda \times_s Mor(C)$ (the fiber product):

$$\array{ && S {}_\lambda \times_s Mor(C) \\ & {}^{pr_1}\swarrow && \searrow^{pr_2} \\ S && && Mor(C) \\ & \searrow^{\lambda}& & {}^{s}\swarrow && \searrow^{t} \\ && Obj(C) &&&& Obj(C) } \,.$$

Above we have seen that the action of $C$ on $S$ sends every element in this fiber product, which is a pair

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

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 $S$:

$$\array{ && S {}_\lambda \times_s Mor(C) \\ & \swarrow && \searrow^{t \circ pr_2} \\ pt &&\downarrow^{\rho}&& Obj(C) \\ & \nwarrow && \nearrow_{\lambda} \\ && S }$$

But $\rho$ satisfies yet another compatibility condition: so far we have only used the source-target mathcing condition of the functor $\rho : C \to Set$. 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 $Mor(C) {}_t \times_s Mor(C)$ is itself the tip of a span arising from composing the span of $C$ with itself by pullback:

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

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

$$\array{ && Mor(C) {}_t \times_s Mor(C) \\ & {}^{s \circ pr_1}\swarrow && \searrow^{t \circ pr_2} \\ Obj(C) &&\downarrow^{\circ}&& Obj(C) \\ & {}^{s}\nwarrow && \nearrow_{t} \\ && Mor(C) } \,.$$

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

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

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 $\rho:C\rightarrow \Cat$ with the expectation, that this then is just a module for $C$ as a monad.