Timeline for Given a map of classifying spaces, can the target be described as a groupoid quotient of the source mod some action of some (co)kernel?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 17, 2015 at 18:50 | vote | accept | Theo Johnson-Freyd | ||
| Apr 15, 2015 at 8:06 | history | edited | Chris Schommer-Pries | CC BY-SA 3.0 |
clearer notation.
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| Apr 15, 2015 at 8:06 | comment | added | Chris Schommer-Pries | For your first question, I just meant the space corresponding to the groupoid $G//(H \times H)$. I see that is confusing and will edit. You can check this for the case $H=1$, where you just get G, the loop space of BG. For your second question I think that this is probably also correct in the algebrogeometric context. I think the easiest way to check is to just check the universal property. A map into the fiber product is a pair of maps to BH and a isomorphism between the pushforwards into BG. But this is exactly what the stack $U_1$ classifies, no? | |
| Apr 14, 2015 at 21:52 | comment | added | Theo Johnson-Freyd | Oh, these formulae certainly make sense for (derived?) algebraic stacks. Do you know if they are known to be "correct" in the algebrogeometric world? E.g. I really do want to work with the algebraic groups G,H qua schemes, and not just work with the topological groups of $\mathbb C$-points. | |
| Apr 14, 2015 at 21:50 | comment | added | Theo Johnson-Freyd | Awesome. I'll have to think a bit to make sure I understand this formula. Is it clear that this double coset groupoid is groupal, so that I can take B of it? Or on the right did you just mean the space corresponding to the groupoid $G // (H \times H)$? | |
| Apr 14, 2015 at 11:22 | history | answered | Chris Schommer-Pries | CC BY-SA 3.0 |