This is a question of terminology. I want to talk about the category whose...


* ...objects are pairs $(G,M)$, where $G$ is a group and $M$ is a $G$-set.
* ...morphisms $(G,M)\rightarrow (G',M')$ are pairs $(f\_G,f\_M)$, where $f\_G:G\rightarrow G'$ is a group homomorphism, and $f\_M:M\rightarrow M'$ is a set map such that
$$ f\_M(g\cdot m) = f\_G(g)\cdot f\_M(m)$$
for all $g\in G$ and $m\in M$.


I've been calling these **decorated groups** (and their morphisms), since they've been arising in connection with decorated local systems. However, I'd prefer a more standard name, hopefully one which evokes the correct idea before explanation.



Is there a 'best' name for a group together with a set it acts on?
How about "actions"? "The category of actions" sounds good to me.


