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~~  **I**n general $X$ is (weakly) homotopy equivalent to the fat geometric realization of $C\_U$ (e.g. see Cor. 4.8 here: <http://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!



When is a topological space the homotopy colimit of an open covering?
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](http://www.math.harvard.edu/~lurie/papers/higheralgebra.pdf) 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](http://www.math.uiuc.edu/K-theory/0528/hypercover.pdf).


