Timeline for Constructing a stack (gerbe) from a connected groupoid
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 15, 2013 at 23:56 | answer | added | David Carchedi | timeline score: 3 | |
| Apr 15, 2013 at 13:48 | vote | accept | Mikhail Borovoi | ||
| Apr 15, 2013 at 7:00 | answer | added | David Roberts♦ | timeline score: 3 | |
| Apr 15, 2013 at 6:47 | comment | added | Mikhail Borovoi | @David: If $X\rtimes G$ is the action groupoid corresponding to an action of a $\Gamma$-group $G$ on a $\Gamma$-set $X$, then I should take the groupoid of principal $G$-bundles dominating $X$. What are principal $\mathcal{G}$-bundles when $\mathcal{G}$ is a groupoid? | |
| Apr 15, 2013 at 6:28 | comment | added | Mikhail Borovoi | @David: Yes, the topology is just surjections. | |
| Apr 15, 2013 at 6:19 | comment | added | David Roberts♦ | What's the topology on the category of finite $\Gamma$-sets? Is it just surjections? | |
| Apr 14, 2013 at 21:24 | comment | added | Simon Henry | One problem with this construction is that : start with X being any $\Gamma$-Set, you can construct a groupoid with $A = X \times X$, this groupoid is suppose to be seen as a trivial one, hence we expect the associated stacks to be also trivial. Be if $X$ doesn't have any $\Gamma$ fixed point this is not the case (for example, $\mathcal{G}({*})$ won't have any point). | |
| Apr 14, 2013 at 15:56 | comment | added | Mikhail Borovoi | @Simon: I want to construct a gerbe starting from a connected $\Gamma$-groupoid, and to describe the cohomology class of this gerbe in terms of my $\Gamma$-groupoid, thereby explicitly relating the paper of Springer on non-abelian $H^2$ in Galois cohomology with the book by Giraud. | |
| Apr 14, 2013 at 15:54 | comment | added | Mikhail Borovoi | @Simon: What you propose looks fine. Why is it not what we want to do?! | |
| Apr 14, 2013 at 14:34 | comment | added | Simon Henry | If you don't give more precision on what you want, it seems to me that the simplest things work. (taking $\mathbb{G}(S)$ to be the groupoid whose object are $\Gamma$-equivariant map from $S$ to $X$ and whose arrow are $\Gamma$-equivariant map from $S$ to $A$, the five strucural map being simply composition, and the functoriality being also given by composition... But this is generally not what we want to do. | |
| Apr 14, 2013 at 13:09 | history | asked | Mikhail Borovoi | CC BY-SA 3.0 |