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 $S$.

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).

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

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.

Last revised on January 21, 2024 at 16:36:46. See the history of this page for a list of all contributions to it.