Suppose that $X$ is a topological space and $\left(U_i \to X\right)$ is an open cover. We can associate to it the Cech diagram of this cover $$C_U:\Delta^{op} \to Top.$$ I know that for many good classes of topological spaces, the homotopy colimit of $C_U$ is $X$ (e.g. for manifolds). How general is this result? Does it hold e.g. for locally contractible spaces? I believe that In general $X$ is (weakly) homotopy equivalent to the fat geometric realization of $C_U$ (e.g. see Cor. 4.8 here: https://arxiv.org/abs/0907.3925)
, but for a general simplicial space, this need not agree with hocolim. Any feedback or references would be appreciated. Thanks!
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$\begingroup$ Quillen's Theorems A/B are relevant, maybe? $\endgroup$– user43326Commented Nov 23, 2014 at 11:01
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2$\begingroup$ 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... $\endgroup$– Marc HoyoisCommented Nov 23, 2014 at 16:45
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$\begingroup$ 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! $\endgroup$– Marc HoyoisCommented Nov 23, 2014 at 17:12
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$\begingroup$ 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". $\endgroup$– Marc HoyoisCommented Nov 24, 2014 at 1:53
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1$\begingroup$ 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). $\endgroup$– Lennart MeierCommented Apr 13, 2017 at 8:17
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2 Answers
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It is true in complete generality that $X$ is the homotopy colimit of $C_U$ (and hence that the fat realization computes the homotopy colimit in this case). This is a special case of Lurie's version of the Seifert-van Kampen theorem. More precisely, Proposition A.3.2 in Higher Algebra says that that the "underlying homotopy type" functor
$$Sing: Open(X) \to \mathcal{S} $$
is a cosheaf, so it lifts to a colimit-preserving functor $Shv(X) \to \mathcal{S}$. Since $X$ is the colimit of $C_U$ in $Shv(X)$, $Sing(X)$ is the colimit of $Sing(C_U)$ in $\mathcal{S}$.
ETA: The proof of the above result actually shows that $Sing$ is a hypercomplete cosheaf. This reminded me that Dugger and Isaksen also prove this fact in their paper Hypercovers in topology (Wayback Machine).
answered Nov 23, 2014 at 17:31
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$\begingroup$ My question mathoverflow.net/questions/102295/… is relevant to your comment. Note that this sometimes called "small simplex" theorem has other proofs in the literature, and has not led to proofs of higher Seifert-van Kampen Theorems, to my knowledge. $\endgroup$ Commented Nov 23, 2014 at 20:42
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$\begingroup$ I think it would be unfair not to call Lurie's theorem itself a higher Seifert-van Kampen theorem. Note that the result I've quoted is only a special case of half of the theorem. The other half (in the same special case) says that the actual colimit of the diagram of simplicial sets $Sing(C_U)$ is weakly equivalent to $Sing(X)$. $\endgroup$ Commented Nov 23, 2014 at 23:03
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$\begingroup$ My attitude to Seifer-van Kampen thoerems is that they are about direct computation of a strict homotopical invariant as an exact colimit. $\endgroup$ Commented Nov 24, 2014 at 15:03
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$\begingroup$ The invariants I have dealt with are of: spaces with many base points; filtered spaces; and $n$-cubes of spaces. The first two are dealt with in the EMS Tract vol 15 (2011) on "Nonabelian algebraic topology:...", and were published first in 1967 and 1981, respectively. $\endgroup$ Commented Nov 24, 2014 at 15:17
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$\begingroup$ The link to Lurie's Higher algebra seems to be dead - perhaps this would be a reasonable replacement: math.ias.edu/~lurie/papers/HA.pdf? I have added Wayback Machine link for the other dead link - but maybe it could be simply replaced by a link to arXiv:math/0111287. $\endgroup$ Commented Jun 19 at 7:58
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Tammo tom Dieck's book "Algebraic topology" (EMS 2008, doi:10.4171/048) has Section 13.2 on the "homotopy colimit of a covering" which you should find relevant.
answered Nov 23, 2014 at 11:23