Sorry for the long post but this is a personal piece of maths, and I needed to be more precise as possible.
There exists a well known equivalence between the category of $G$-sets and the category of functors $Fun(G,\mathbf{Sets})$ (viewing $G$ as a category with a single object $*$): given a set $X$ just consider the unique functor sending $*$ into $X$; functoriality determines the well known classical permutation representation $G\to Sym(X)$, which easily leads to the equivalent notion of an action as a map $G\times X\to X$ with suitable properties.
It is also well known that the notion of "action" of a "group object" can be stated in any category with finite products and a terminal object (say, $1$).
I would like to relate the two notions: I started noticing that the first relation lacks of elasticity, being formulated in the particular case of the category of sets. My first task is hence to generalize the notion of action of a group on some $X$, object in $\mathbf C$ (category w. finite products and a terminal object). It seems to me that I can define it as a functor $G\to \mathbf C$ sending $*$ into $X$: Functoriality allow me to think that any $g\in G$, seen as an isomorphism $g\colon *\to *$, corresponds via $F$ to an isomorphism in $\mathbf C$, then $G$ is related to a subset(subgroup) in $\hom_\mathbf C(X,X)$.
So, it seems we recovered the notion of permutation representation in this more general context. Futhermore, it seems to me that when we consider, say, a functor $F\colon G\to \mathbf{Top}$ ($G$ a group in that category, i.e. a topological group), mere functoriality allows me to say that $G\times X\to X$ is a continouos map (similarly consider then the subcategory of $\mathbf{Top}$ made by differentiable manifolds, $G$ is then a Lie group and the action a smooth map).
Can we go any further? Moerdijk defines the action of a groupoid on a space $\mathbf G=(s,t:G_\text{mor} \to G_\text{ob})$ as an arrow $\mu\colon G_\text{mor}\times_{G_\text{ob}} E\to E$ with suitable properties. This is partly similar to an idea I got yesterday, thinking to a suitable way to expand the notion of $G$ acting on something.
In a few words, I start identifying $G$ and $\hom(*,*)$, groups in set-theoretic sense wrt the composition. Say that an action of $G$ on an object $X$ in $\mathbf C$ is a function $\hom(*,*)\times \hom(1,X)\to \hom(1,X)$ which is an action in the classical set-theoretic sense: is it formally correct? Can this approach be applied in a "real case"? Can you provide me a reference for the definition I gave of Moerdijk groupoid-action, which I read dunno-where on MO?
Thanks everybody.