Timeline for Is there a 'best' name for a group together with a set it acts on?
Current License: CC BY-SA 3.0
12 events
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| May 13, 2011 at 8:28 | comment | added | TaQ | Also, the neutral element $e$ of the group is the unique $e$ with $(e,e,e)\in g$ . Here it is understood that one defines $(x,y,z)=((x,y),z)$ . | |
| May 13, 2011 at 8:28 | comment | added | TaQ | Formally and irredundantly, a group action would be a pair $A=(g,f)$ , or equivalently (provided that $M$ is not empty) the function $(u,v,x)\mapsto(g(u,v),f(u,x))$ , where $g$ is the group operation $G\times G\to G$ and $f$ is the "action" $G\times M\to M$ . Since any function $h$ uniquely determines its domain ${\rm dom\ }h=\{w:\exists z;(w,z)\in h\}$ , the pair $A$ , and already even the function $f$ then uniquely determines both the underlying set $G$ of the "group" and the set $M$ on which "the group acts". | |
| May 12, 2011 at 23:06 | answer | added | Theo Johnson-Freyd | timeline score: 2 | |
| May 12, 2011 at 22:35 | comment | added | Neil Strickland | Such a pair $(G,M)$ gives a groupoid $M//G$ with object set $M$, and one morphism from $x$ to $y$ for each $g\in G$ with $gx=y$. Every morphism $(G,M)\to (G',M')$ gives a functor $M//G\to M'//G'$. Most things that you might want to do can be made to work for general groupoids, and that extra generality is often useful, as well as sidestepping the problem of terminology. | |
| May 12, 2011 at 20:20 | comment | added | Martin Brandenburg | These types of categories appear in the literature (in different areas of universal algebra and category theory), and just because you haven't seen it so much does not mean that it is not natural to consider them. | |
| May 12, 2011 at 19:54 | comment | added | Charles Staats | There are other similar categories--for instance, the category with pairs (ring R, R-module) as objects, in which a morphism $(R, M) \to (R', M')$ is a ring morphism $R \to R'$, together with an $R$-module morphism $M \to M'$. It would be nice to have a similar naming conventions for all of these. Perhaps these categories turn out to be less useful than one might suppose (e.g., the category I just described is not abelian), and so they don't make it into very many published works, hence don't have well-known names. | |
| May 12, 2011 at 17:21 | comment | added | Jim Humphreys |
Most of the time people tend to fix a specific group $G$ and then refer to "the category of $G$-sets" (with obvious morphisms). What is wanted here seems to come up rarely in practice: letting the group as well as the set vary arbitrarily. I'd be content just to refer to the category as before, but with no specific $G$ fixed. I also have no objection to "the category of group actions" (whereas "the category of actions" is really too vague). Until mathematicians elect a dictator to decide such things, we'll probably all do our own thing anyway.
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| May 12, 2011 at 17:16 | vote | accept | Greg Muller | ||
| May 12, 2011 at 17:05 | comment | added | Jorge Vitório Pereira | If you replace group by semi-group your objects are what some call dynamical systems and your morphisms are usually called semi--conjugacies. To some "the category of invertible dynamical systems" would evoke the right idea, but "the category of actions" as proposed by Angelo sounds much better to me. | |
| May 12, 2011 at 16:47 | answer | added | Angelo | timeline score: 13 | |
| May 12, 2011 at 16:44 | comment | added | Noah Snyder | The category of group-sets? | |
| May 12, 2011 at 16:42 | history | asked | Greg Muller | CC BY-SA 3.0 |