Suppose $X$ is a (paracompact, Hausdorff) topological space and we want to define its Cech cohomology with coefficients in $\mathbb Z$. Here is the way I have seen this constructed. For each open cover $\mathcal U$, we define the complex ${\check{\mathcal C}}^\bullet(\mathcal U,\mathbb Z)$ of Cech cochains relative the open cover with the usual Cech differential. Now, if $\mathcal V$ is another open cover of $X$ which is a refinement of $\mathcal U$, then we have a map ${\check{\mathcal C}}^\bullet(\mathcal U,\mathbb Z)\to {\check{\mathcal C}}^\bullet(\mathcal V,\mathbb Z)$ for each "presentation" of $\mathcal V$ as a refinement of $\mathcal U$. But the chain homotopy class of this map is well-defined and thus, we get a well-defined directed system of groups $\check{H}^\bullet(\mathcal U,\mathbb Z)$ indexed by the open covers $\mathcal U$ of $X$, and their direct limit is defined to be $\check{H}^\bullet(X,\mathbb Z)$. On the other hand I have heard people speak of the complex $\check{\mathcal C}^\bullet(X,\mathbb Z)$ of Cech cochains on a space $X$, which computes the Cech cohomology of $X$.


I am interested in knowing the definition of this complex which computes the Cech cohomology. It can't simply be the direct limit of the complexes ${\check{\mathcal C}}^\bullet(\mathcal U,\mathbb Z)$ since these do not form a directed system of complexes (as the maps are well-defined only up to homotopy).



Is there a complex which computes Cech cohomology?
This is the kind of thing sieves are good for. For an open cover, let $S$ denote the sieve it generates, so $S$ is poset of open subsets $V$ such that $V$ is contained in some element of the cover. A quasi-isomorphic model for the Cech complex is given by the "homotopy limit over the sieve", so the $0$th term in the complex is the product of the $F(U)$ for all $U$ in $S$, the $1$st term is the product of the $F(U)$ for all $U \subset U'$ in $S$, etc. Basically, instead of modeling "a section on each element of the open cover, which agree on intersections", you model "a section on each element of the sieve, which agree on restriction". The point now is that a refinement of open covers gives an honest inclusion of sieves, and the sieve model for the Cech complex is just plain functorial in inclusions of sieves. So you can just pass to the filtered colimit.


