Refinement-colimit Čech cohomology

Assume the Axiom of Choice (The Axiom of Choice). Let $X$ be a topological space and let $\mathcal{F}$ be a sheaf of abelian groups on $X$. The topology ${\mathcal{T}}_X$ is a set of open subsets. Fix a well-order of ${\mathcal{T}}_X$ using the Axiom of Choice, and let $\mathrm{Cov}(X)$ consist of all subsets $\mathcal{U}\subseteq {\mathcal{T}}_X$ whose union is $X$, each indexed by its distinct members in the inherited well-order. This is a set of ordered covers. An arbitrary indexed open cover is represented by its distinct-member cover: under Choice, one may choose an index for each distinct member, so the two cover presentations refine one another and yield canonically isomorphic Čech cohomology under refinement independence. For $\mathcal{U}\in \mathrm{Cov}(X)$ write ${\check{H}}^p(\mathcal{U},\mathcal{F})$ for its fixed-cover Čech cohomology (Fixed-cover Čech cohomology).

Say that $\mathcal{U}⪯\mathcal{V}$, or that $\mathcal{V}$ refines $\mathcal{U}$, when there is a refinement function from $\mathcal{V}$ to $\mathcal{U}$, that is, a map $c$ from the index set of $\mathcal{V}$ to the index set of $\mathcal{U}$ with $V\subseteq c(V)$ for every $V\in \mathcal{V}$ (Refinement map of ordered open covers). This relation is a preorder: it is reflexive through the identity map, and it is transitive because a composite of refinement functions is a refinement function. It is directed: for covers $\mathcal{U},\mathcal{V}\in \mathrm{Cov}(X)$ the set of distinct intersections $\mathcal{W}=\{U\cap V:U\in \mathcal{U},V\in \mathcal{V}\}$ is another member of $\mathrm{Cov}(X)$, indexed by the same inherited well-order, and covers $X$. For each $W\in \mathcal{W}$, choose the least $U\in \mathcal{U}$ and the least $V\in \mathcal{V}$ with $W=U\cap V$; the resulting maps refine $\mathcal{W}$ to both covers. Repeating this construction stays within the same set $\mathrm{Cov}(X)$.

The Axiom of Choice is used to fix transition data: for every pair $\mathcal{U}⪯\mathcal{V}$ choose one refinement function $c_{\mathcal{V}\mathcal{U}}$ from $\mathcal{V}$ to $\mathcal{U}$. By Refinement choices induce the same Čech map any two refinement functions from $\mathcal{V}$ to $\mathcal{U}$ induce the same homomorphism ${\check{H}}^p(\mathcal{U},\mathcal{F})\to {\check{H}}^p(\mathcal{V},\mathcal{F})$ in every degree, so the chosen data give well-defined maps ${\check{H}}^p(\mathcal{U},\mathcal{F})⟶{\check{H}}^p(\mathcal{V},\mathcal{F})(\mathcal{U}⪯\mathcal{V}),$ and these maps are compatible with composition and with identities: the composite of the chosen refinement functions along $\mathcal{U}⪯\mathcal{V}⪯\mathcal{W}$ is again a refinement function from $\mathcal{W}$ to $\mathcal{U}$ and hence induces the composite of the two induced maps, while the identity is a refinement function of a cover to itself and induces the identity. Thus $\mathcal{U}\mapsto {\check{H}}^p(\mathcal{U},\mathcal{F})$ is a functor from the directed preorder $\mathrm{Cov}(X)$ to the category of abelian groups.

The Čech cohomology of $X$ with values in $\mathcal{F}$ is the filtered colimit taken in the category of abelian groups, of that functor (Filtered categories and filtered colimits). A class of ${\check{H}}^p(X,\mathcal{F})$ is represented by a pair $(\mathcal{U},\alpha )$ with $\mathcal{U}\in \mathrm{Cov}(X)$ and $\alpha \in {\check{H}}^p(\mathcal{U},\mathcal{F})$, and two such pairs represent the same class exactly when the two classes agree after refinement to a common cover. A morphism $\phi :\mathcal{F}\to \mathcal{G}$ of abelian sheaves induces the maps ${\check{H}}^p(\mathcal{U},\phi )$ on fixed covers, which commute with the refinement maps because the cochain maps of Refinement map of ordered open covers are defined componentwise from the values of the cochains; these maps pass to the filtered colimit and give ${\check{H}}^p(X,\phi ):{\check{H}}^p(X,\mathcal{F})⟶{\check{H}}^p(X,\mathcal{G}),$ so that ${\check{H}}^p(X,-)$ is a functor on the abelian sheaves on $X$ with ${\check{H}}^p(X,\mathcal{F})=0$ for $p<0$.