I am currently trying to understand Cech cohomology. Five questions arised and I would be glad for help. In what follows $X$ is a topological space.

I really like Dugger's and Isaksen's paper "Topological Hypercovers and ...". They prove for an arbitrary open cover $U=(U_a)_{a\in A}$ of $X$ the weak equivalence $$ \operatorname{hocolim} ~C(U)\to X $$ with $C(U)$ is the usual simplicial space of Cech namely $$ \dots\to\coprod_{(b,c)\in A\times A}U_b\cap U_c\to\coprod_{a\in A}U_a $$ Related statements are due to Segal. If $U$ is now a cover with every iterated intersection contractible we have a weak equivalence $$ \operatorname{hocolim} ~C(U)\to \operatorname{hocolim} ~\pi_0C(U) $$ The space $\pi_0C(U)$ is a simplicial set. You can find such a cover for every locally contractible space for instance. Dugger and Isaksen build the category $OpCov(X)$. This has open covers of $X$ as objects and morphisms are mappings between the index sets $f:A\to B$ and for every $a$ in $A$ a mapping $U_a\to V_{f(a)}$. One can also build the category $Cov(X)$ with same objects but morphisms only strict containments of covers. Is then $$ \operatorname{hocolim} ~C(U)\sim \lim_{U\in Cov(X)}~\operatorname{hocolim} ~\pi_0C(U)\sim \lim_{U\in OpCov(X)}~\operatorname{hocolim} ~\pi_0C(U) $$ all weakly equivalent for locally contractible $X$?

Why does one define Cech cohomology as $H^n(X)=\operatorname{colim} H^n(\operatorname{hocolim} C(U))$ with colimit over $Cov(X)^{op}$? Is this the right definition? Why not the limit over $Cov(X)$ instead of colimit over the opposite category?

What is the problem with Cech Homology? One defines $\hat H^n(X)=\operatorname{colim} ~H^n(C(U))$ as Cech cohomology. Then for a locally contractible $X$ it coincides with singular cohomology. Why not define $\hat H_n(X)=\lim ~H_n(C(U))$. I have read that it is because the limit functor does not respect exact sequences. Does it mean $\hat H_n(X)$ coincides

**not**with singular homology for locally contractible $X$?Can one see directly without going through sheaf cohomology that singular cohomology and Cech cohomology are the same for locally contractible $X$? I have written down complexes for the boundary of the two simplex with a nice cover and yes the cohomology is the same. Is there a kind of MayerVietoris argument?

Here is my last question. $X$ is a good space now. Where is the mistake in the following equality. I consider Top as a topologgical model category and the crucial step is perhaps the one which looks like a smallness argument on the homotopy category. $$ \begin{array}{rcl} \pi_0(X)&=&[S^0, \operatorname{hocolim} ~C(U)]\\ &=& \operatorname{hocolim}[S^0, C(U)]\\ &=& \operatorname{hocolim} \pi_0(C(U))\\&=&X \end{array} $$ Here $U$ is a cover with everything contractible as above.

Thank endurance reading and for help.

`$\pi_0 C(U)$`

, but this can have higher homomtopy. Try working it out explicitly for a cover of the circle. (Also, since you are working with unpointed hocolim, $S^0$ should perhaps be replaced with a single point.) $\endgroup$