Action of group object

Question

Let $C$ be a category with finite products and $G$ a group object in $C$. How to formalize an action of $G$ on an other object $X$ of $C$.

Thanks in advance for your help.

Answer 1

You can just rewrite standard definitions of group action, but in some general category $\mathcal C$, action won't be a function, but a morphism. Thus, let $G$ be group object with multiplication $m\colon G\times G\to G$ and unit $e\colon 1\to G$. Let $a\colon G\times X\to X$ be a morphism such that

\begin{align}a\circ(e\times\operatorname{id}) &= \operatorname{id},\\ a\circ(m\times\operatorname{id}) &= a\circ(\operatorname{id}\times a).\end{align}

If you have elements, you can easily see that these are the usual axioms.

A bit more troublesome would be to define action as $a\colon G\to \operatorname{Hom}(X,X)$ if $\mathcal C$ is not enriched over itself, i.e. $\operatorname{Hom}(X,X)$ is object of $\mathcal C$. Perhaps, one could use some trick to overcome this, but I'm not seeing it immediately.