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).

canonically definedcomplex computing the Cech cohomology in the general case (independent of the choice of any acylic cover). $\endgroup$ – Mohan Swaminathan Nov 4 '17 at 3:521more comments