Is there a 'best' name for a group together with a set it acts on? - MathOverflow

Is there a 'best' name for a group together with a set it acts on?

2011-05-12T16:42:43Z

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.

Answer by Angelo for Is there a 'best' name for a group together with a set it acts on?

2011-05-12T16:47:01Z

How about "actions"? "The category of actions" sounds good to me.

Answer by Theo Johnson-Freyd for Is there a 'best' name for a group together with a set it acts on?

2011-05-12T23:06:39Z

Angelo's answer above is clearly the correct one. But, somewhat tongue-in-cheek, I would also like to recommend "The category of trivialized groupoids". As I'm sure you're very aware, there's a pretty good analogy

vector bundles : trivialized vector bundles :: groupoids : group actions.