Timeline for universal families and maps to quotient stacks
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 11, 2013 at 16:16 | comment | added | IMeasy | Yes, I should have said canonical. Universal has a categorical meaning which is not true here. Thank you! | |
| Apr 11, 2013 at 16:01 | comment | added | David Carchedi | By the way, in some more detail, there is a canonical map $$\left[X//G\right] \to \left[pt//G\right]=BG$$, which is faithful. This is how you get a $G$-torsor out of a map $$S \to \left[X//G\right]$$ (it is classified by the composition into $\left[pt//G\right].$) I should be a bit careful, since I am used to working with topological stacks, but I think everything should work. | |
| Apr 11, 2013 at 15:57 | comment | added | David Carchedi | What do you mean by "universal"? To me, universal means that every family is a pullback of it, which is not true here, but is locally- that's why I said locally universal. If instead, you mean canonical, then sure. | |
| Apr 11, 2013 at 15:18 | comment | added | IMeasy | But I still don't understand one thing. By pulling back the universal family from $[X//G]$ to $X$, don't I get a "universal" $G$-invariant family on $X$? | |
| Apr 11, 2013 at 15:13 | comment | added | IMeasy | Ok let's say that I composed with a forgetful functor $f: Groupoids \to Sets$.... :) Just kidding, it is a good remark, I edited. That's more or less what I felt: that I can lift the map $S$-globally iff the torsor $X \to [X//G]$ is trivial over the image of $S$. | |
| Apr 11, 2013 at 15:05 | vote | accept | IMeasy | ||
| Apr 11, 2013 at 15:05 | vote | accept | IMeasy | ||
| Apr 11, 2013 at 15:05 | |||||
| Apr 11, 2013 at 13:24 | history | answered | David Carchedi | CC BY-SA 3.0 |