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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 \[ hocolim ~C(U)\to X \] with $C(U)$ is the usual simplicial space of Cech namely \[ ...\to\coprod_{(b,c)\in A\times A}U_b\cap U_c\to\coprod_{a\in A}U_a \] Related statements are due to Segal. If $U$ is now a cover with every iterated intersection contractible we have a weak equivalence \[ hocolim ~C(U)\to hocolim ~\pi_0C(U) \] 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 \[ hocolim ~C(U)\sim lim_{U\in Cov(X)}~hocolim ~\pi_0C(U)\sim lim_{U\in OpCov(X)}~hocolim ~\pi_0C(U) \] all weakly equivalent for locally contractible $X$?

  • Why does one define Cech cohomology as $H^n(X)=colim H^n(hocolim C(U))$ 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 $\hat H^n(X)=colim ~H^n(C(U))$ 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 $X$?

  • Singular cohomology is homotopy classes of maps in EilenbergMclane space \[ [X,K(n)]=[hocolim ~C(U),K(n)]=[lim_{U\in Cov(X)}~hocolim ~\pi_0C(U),K(n)] \] Of the last equality I am not sure as explained before. Now one can take limit out. \[ [X,K(n)]=colim[hocolim ~\pi_0C(U),K(n)] \] I am not familiar with homotopy colimits. Can one take it out too?

  • 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. \[ \begin{array}{rcl} \pi_0(X)&=&[S^0, hocolim ~C(U)]\\ &=& hocolim[S^0, C(U)]\\ &=& hocolim \pi_0(C(U))\\&=&X \end{array} \] Here $U$ is a cover with everything contractible as above.

Thank endurance reading and for help.
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Understand Cech Cohomology

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 \[ hocolim ~C(U)\to X \] with $C(U)$ is the usual simplicial space of Cech namely \[ ...\to\coprod_{(b,c)\in A\times A}U_b\cap U_c\to\coprod_{a\in A}U_a \] Related statements are due to Segal. If $U$ is now a cover with every iterated intersection contractible we have a weak equivalence \[ hocolim ~C(U)\to hocolim ~\pi_0C(U) \] 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 \[ hocolim ~C(U)\sim lim_{U\in Cov(X)}~hocolim ~\pi_0C(U)\sim lim_{U\in OpCov(X)}~hocolim ~\pi_0C(U) \] all weakly equivalent for locally contractible $X$?

  • What is the problem with Cech Homology? One defines $\hat H^n(X)=colim ~H^n(C(U))$ 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 $X$?

  • Singular cohomology is homotopy classes of maps in EilenbergMclane space \[ [X,K(n)]=[hocolim ~C(U),K(n)]=[lim_{U\in Cov(X)}~hocolim ~\pi_0C(U),K(n)] \] Of the last equality I am not sure as explained before. Now one can take limit out. \[ [X,K(n)]=colim[hocolim ~\pi_0C(U),K(n)] \] I am not familiar with homotopy colimits. Can one take it out too?

  • 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. \[ \begin{array}{rcl} \pi_0(X)&=&[S^0, hocolim ~C(U)]\\ &=& hocolim[S^0, C(U)]\\ &=& hocolim \pi_0(C(U))\\&=&X \end{array} \] Here $U$ is a cover with everything contractible as above.

Thank endurance reading and for help.