Timeline for The contravariant mapping space represented by a homotopical classifying space (e.g. BG)
Current License: CC BY-SA 4.0
15 events
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| Dec 14, 2021 at 10:59 | comment | added | Denis Nardin | Said it differently, the definition of $BG$ is given via the geometric realization of the bar construction, and this clearly commutes with pullbacks. So if $G$ is a constant sheaf, so is $BG$ | |
| Dec 14, 2021 at 6:57 | comment | added | Denis Nardin | @DmitriPavlov That is true basically by definition - with $BG$ I'm indicating $|B \operatorname{Sing}(G)|$, since this is the object produced by the recognition theorem (replacing $G$ with $\operatorname{Sing}(G)$ is allowed since $G$ has the homotopy type of a CW complex) | |
| Dec 14, 2021 at 6:20 | comment | added | Dmitri Pavlov | It seems that the (first part of) Remark A.1.4 needs BG to be homotopy equivalent to a CW-complex in order for the constant sheaf on BG to be representable by the space BG. Assuming G is a CW-complex, how do you show that BG is homotopy equivalent to a CW-complex in this generality? | |
| Dec 13, 2021 at 10:23 | history | edited | Denis Nardin | CC BY-SA 4.0 |
Added an explanation for why constant sheaf
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| Dec 13, 2021 at 10:13 | history | edited | Denis Nardin | CC BY-SA 4.0 |
added 125 characters in body
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| Dec 13, 2021 at 10:12 | comment | added | Denis Nardin | by a topological space in the general case (at least if $G$ is Hausdorff), but this requires a lot more care than I am using in this answer (essentially one needs to carefully see how the geometric realization interacts with the functor from topological spaces to sheaves on paracompact Hausdorff spaces). For example, what should $BG$ be for $G$ a profinite group? | |
| Dec 13, 2021 at 10:11 | comment | added | Denis Nardin | @DmitriPavlov I'll add more details, but I did not mean to literally apply the results of remark A.1.4 (that does require an additional hypothesis), but merely the first sentence, giving a description of the constant sheaves. Precisely this follows from a combination of 7.1.4.4 and 7.1.5.1 in Higher Topos Theory (it is mildly frustrating that HTT does not contain the statement in a citeable form). But you are right that something funny is happening: the statement I am making is correct only when $G$ has the homotopy type of a CW complex. I still believe the sheaf $BG$ is representable (cont.) | |
| Dec 12, 2021 at 7:41 | comment | added | Dmitri Pavlov | Remark A.1.4 requires an additional condition from X, it is not applicable to arbitrary paracompact Hausdorff topological spaces. | |
| Dec 12, 2021 at 6:59 | comment | added | Denis Nardin | @DmitriPavlov that's again Remark A.1.4 in Higher algebra, using the fact that $X$ is paracompact (precisely one should restrict to the basis of open $F_\sigma$ subsets of $X$, as in HTT.7.1.1) | |
| Dec 11, 2021 at 15:12 | comment | added | Dmitri Pavlov | Yes, so why is the sheaf with the n-simplices as indicated isomorphic to the constant sheaf on G (or rather Sing(G))? | |
| Dec 11, 2021 at 11:07 | comment | added | Denis Nardin | @DmitriPavlov It's the automorphism sheaf of the trivial $G$-bundle, which can be checked to be $G$ ($n$-simplices over $U$ are $G$-equivariant maps $G\times U\times \Delta^n\to G\times U$ over $U$, that is continuous maps $U\times\Delta^n\to G$) | |
| Dec 10, 2021 at 16:46 | comment | added | Dmitri Pavlov | Why is ΩBun_G(−) the constant sheaf on G? | |
| Nov 24, 2021 at 18:00 | vote | accept | Doron Grossman-Naples | ||
| Nov 23, 2021 at 9:52 | history | edited | Denis Nardin | CC BY-SA 4.0 |
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| Nov 23, 2021 at 9:13 | history | answered | Denis Nardin | CC BY-SA 4.0 |