Timeline for Is there a description of sheaf cohomology in algebraic-topological terms?
Current License: CC BY-SA 2.5
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 9, 2010 at 21:03 | comment | added | Chris Schommer-Pries | That sounds like it should work if all the spaces are reasonable, but there are technical pitfalls when working in the relative setting. What you want is another abelian group object $E \to X$ with an embedding as a closed sub-object $Y \hookrightarrow E$ such that E is "contractible" in the over category, i.e. $E \simeq X$ as spaces over X. Then the quotient E/Y in spaces over X will be the correct group BY. If you phrase it like this, then most (all?) of Segal's machinery should carry over. | |
| Mar 9, 2010 at 3:11 | comment | added | Omar Antolín-Camarena | Right, I knew something like this must be true for constant coefficients (but not a precise set of hypothesis and a reference). Could you explain to me explicitly the case with arbitrary Y? Given an Abelian group object Y in {spaces over X}, to define BY as a space over X do you just form the (simplicial space over X valued) nerve of the Abelian group object $Y\to X$, and then take its realization (defined by the usual nerve and realization adjunction for the functor $\Delta \to$ {spaces over X} that sends the n-simplex to the projection $|\Delta^n|\times X \to X$)? | |
| Mar 9, 2010 at 2:38 | history | edited | Chris Schommer-Pries | CC BY-SA 2.5 |
clarified statement about when BA is a group.
|
| Mar 9, 2010 at 1:02 | history | answered | Chris Schommer-Pries | CC BY-SA 2.5 |