Functors inducing group actions.

Question

I'm reading Riehl and I have a question about one of her examples. The setup is as follows.

Suppose that $G$ is a group, which we can consider as a one object category $BG$ with formal object $X$. If $F: BG \rightarrow \mathscr{C}$ is a functor, then there are a few things we can note.

1. Since there is only a single object $X$ in $BG$, every morphism $g:X \rightarrow X$ in $BG$ is taken to an endomorphism $g^\ast:FX \rightarrow FX$ in $\mathscr{C}$. Stronger than this, since every morphism in $BG$ is an isomorphism, the elements of $BG$ are actually sent to automorphisms of $FX$.
2. The functoriality axioms imply two things. Firstly that if $g,h:X \rightrightarrows X$, then $(gh)^\ast = g^\ast h^\ast$. Secondly that $(1_X)^\ast = 1_{FX}$.

She then writes that the functor $F:BG \rightarrow \mathscr{C}$ defines an action of $G$ on the object $FX \in\mathscr{C}$.

I've only ever encountered the notion of a group acting on a set, so I want to make sure I'm understanding what's going on here. In the case where $F:BG \rightarrow \mathbf{Set}$, then the functor associates any $G \ni g \mapsto g^* \in S(FX)$, i.e it sends a group element to an element of the symmetric group on $FX$. Now I know that if you have a group homomorphism $\phi_g:G \rightarrow S(FX)$ then you have a group action by defining $g \ast a = \phi_g(a)$. Are the functoriality axioms telling us that the functor $F$ is in some sense a group homomorphism from $G$ to $S(FX)$ and in that way $F$ induces a group action?

Thanks in advance for the help. Let me know if I can clarify.

Answer 1

Given any object $c$ in a category $\mathbf{C}$ (with a set of automorphisms), one can show that $\text{Aut}_\mathbf{C}(c)$ is a group with composition as its operation. A group $G$ acting on $c$ is a group homomorphism $\varphi\colon G\to\text{Aut}_\mathbf{C}(c)$, and a functor $F\colon BG\to\mathbf{C}$ induces such a homomorphism $\varphi(g) = F(g)$ (check if you doubt it), when $c=F(\text{the unique object of }BG)$. In this sense, it makes sense to define the group's action as a functor.