Timeline for What is the standard notation for group action
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 11, 2011 at 7:01 | comment | added | Pete L. Clark | Upon closer reading I see that Theo has more general objects than abstract groups (or monoids) doing the "acting", so my comment was not especially pertinent. I have deleted it. | |
| May 11, 2011 at 6:57 | comment | added | Pete L. Clark | But I take your point nevertheless: the fact that e.g. $\operatorname{Top}$ is not Cartesian closed causes problems in the definition of topological group actions. | |
| May 11, 2011 at 6:50 | comment | added | Pete L. Clark | @Harry: A topological group is not (just) a group, and a group scheme is definitely not a group. So your examples lie outside of my philosophy. Perhaps we could talk about "enriched categories"...but I'd rather not. | |
| May 11, 2011 at 1:42 | comment | added | Theo Johnson-Freyd | @Ben: OP makes a similar point. @Scott: There are situations where I have a reasonable notion of "acts on", which is more complicated than a map $G \times X \to X$; e.g. if $G$ is a (Hopf) algebra and $X$ a representation, then I certainly don't mean categorical $\times$. But more generally it would be better to write $G \to \operatorname{End}(X)$ than $G \times X \to X$, because then it at least reminds one that the map should be a homomorphism of something. In some settings, though, I think it is not always known what the correct definition of "$\operatorname{End}$" is, e.g. $E_n$ algebras. | |
| May 10, 2011 at 23:55 | comment | added | Mariano Suárez-Álvarez | Scott, that gives you a notation for the action itself, but the proposed notations mean «$G$ acts on $X$», and just expresses the fact that there is an action. | |
| May 10, 2011 at 23:48 | comment | added | Scott Carter | How about $\alpha:G\timesX\rightarrow X$, with $\alpha(g,x)= g\star x$? Of course, I would prefer $\alpha(g,x)= g \cdot x$, and an articulation: $1\cdot x=x$, (gh)\cdot x= g \cdot(h\cdot x)$. Similar notation holds for right actions. | |
| May 10, 2011 at 23:17 | comment | added | Ben Webster♦ | Allen Knutson once pointed out to me that you should always use (1) instead of (2) because in many people's handwriting, the arrow looks a lot like a $G$. | |
| May 10, 2011 at 22:50 | history | answered | Theo Johnson-Freyd | CC BY-SA 3.0 |