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: 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!

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!

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 homotopy equivalent to the fat geometric realization of $C_U$, but for a general simplicial space, this need not agree with hocolim. Any feedback or references would be appreciated. Thanks!

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: 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!