Timeline for When is a topological space the homotopy colimit of an open covering?
Current License: CC BY-SA 4.0
11 events
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| Jun 19 at 8:01 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
http -> https (the question was bumped anyway)
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| Apr 13, 2017 at 8:17 | comment | added | Lennart Meier | The answer probably depends on the chosen weak equivalences in $\mathrm{Top}$, right? For homotopy equivalences you probably need a condition like numerable (see the reference to tom Dieck's book in Ronnie Brown's answer). In the same book, Thm 6.7.11 essentially shows that the same thing holds in complete generality for weak homotopy equivalences (as in Marc's answer, but in more classical language). | |
| Nov 24, 2014 at 1:53 | comment | added | Marc Hoyois | As a follow-up to my first comment, the proof of Lemma A.4.14 in HA shows the following. If $X$ is locally contractible, then $Map(Sh(X),K)\to Map(Sing(X),K)$ is an equivalence provided that the constant sheaf with fiber $K$ is hypercomplete. In particular, $Sing(X)$ and $Sh(X)$ have the same pro-truncated reflections. I suspect they're not the same in general because the usual definition of "locally contractible" for a space corresponds to the topos being "locally $\infty$-connective" rather than actually "locally contractible". | |
| Nov 23, 2014 at 22:36 | vote | accept | David Carchedi | ||
| Nov 23, 2014 at 17:31 | answer | added | Marc Hoyois | timeline score: 22 | |
| Nov 23, 2014 at 17:12 | comment | added | Marc Hoyois | Well, not only is the shape homotopy invariant (Higher Algebra, A.2.10), but your question is actually answered completely in that appendix, see Remark A.3.8: the functor $Shv(X)\to \infty Gpd$ induced by the "underlying homotopy type" functor preserves all colimits! | |
| Nov 23, 2014 at 16:45 | comment | added | Marc Hoyois | This is true in particular when the topos-theoretic shape functor $Sh$ agrees with the classical "underlying homotopy type" functor, since it's always true that $Sh(X)$ is the colimit of $Sh(C_U)$. Lurie shows that $Sh(X)$ is weakly equivalent to $X$ when $X$ is paracompact and homotopy equivalent to a CW complex (Higher Algebra, A.1.4). I'm not sure if you can generalize this to "locally contractible". It's not even clear to me that the shape of a contractible space is contractible... | |
| Nov 23, 2014 at 11:23 | answer | added | Ronnie Brown | timeline score: 8 | |
| Nov 23, 2014 at 11:01 | comment | added | user43326 | Quillen's Theorems A/B are relevant, maybe? | |
| Nov 23, 2014 at 9:46 | history | edited | David Carchedi | CC BY-SA 3.0 |
forgot that I proved something
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| Nov 23, 2014 at 4:15 | history | asked | David Carchedi | CC BY-SA 3.0 |