Is there a 'best' name for a group together with a set it acts on? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T05:27:27Z http://mathoverflow.net/feeds/question/64812 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/64812/is-there-a-best-name-for-a-group-together-with-a-set-it-acts-on Is there a 'best' name for a group together with a set it acts on? Greg Muller 2011-05-12T16:42:43Z 2011-05-12T23:06:39Z <p>This is a question of terminology. I want to talk about the category whose...</p> <ul> <li>...objects are pairs $(G,M)$, where $G$ is a group and $M$ is a $G$-set.</li> <li>...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 $m\in M$.</li> </ul> <p>I've been calling these <strong>decorated groups</strong> (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.</p> http://mathoverflow.net/questions/64812/is-there-a-best-name-for-a-group-together-with-a-set-it-acts-on/64813#64813 Answer by Angelo for Is there a 'best' name for a group together with a set it acts on? Angelo 2011-05-12T16:47:01Z 2011-05-12T16:47:01Z <p>How about "actions"? "The category of actions" sounds good to me.</p> http://mathoverflow.net/questions/64812/is-there-a-best-name-for-a-group-together-with-a-set-it-acts-on/64849#64849 Answer by Theo Johnson-Freyd for Is there a 'best' name for a group together with a set it acts on? Theo Johnson-Freyd 2011-05-12T23:06:39Z 2011-05-12T23:06:39Z <p>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</p> <blockquote> <p>vector bundles : trivialized vector bundles :: groupoids : group actions.</p> </blockquote>