Refinement map of ordered open covers

Let $X$ be a topological space, let $\mathcal{F}$ be a sheaf of abelian groups on $X$ (A sheaf on a topological space), and let $\mathcal{U}=(U_i{)}_{i\in I}$ and $\mathcal{V}=(V_j{)}_{j\in J}$ be open covers of $X$ indexed by linearly ordered sets, with ordered Čech cochains $C^{\bullet }(\mathcal{U},\mathcal{F})$ and $C^{\bullet }(\mathcal{V},\mathcal{F})$ (Ordered Čech cochain complex of a cover) and with the alternating models ${\tilde{C}}^{\bullet }$ of both complexes (Ordered and alternating Čech complexes agree).

A refinement function from $\mathcal{V}$ to $\mathcal{U}$ is a map $c:J\to I$ of the index sets such that $V_j\subseteq U_{c(j)}\text{for every }j\in J\mathrm{.}$ When such a map exists one says that $\mathcal{V}$ refines $\mathcal{U}$ via $c$, or that $\mathcal{V}$ is a refinement of $\mathcal{U}$. A refinement function need not be injective, surjective or order preserving; only the containments $V_j\subseteq U_{c(j)}$ are required, so the members of $\mathcal{V}$ may be assigned to members of $\mathcal{U}$ in any order. A refinement function with $\mathcal{V}=\mathcal{U}$ and $c={\mathrm{id}}_I$ always exists, so every cover refines itself.

Given a refinement function $c$, its Čech cochain map is the cochain map $c^{\mathrm{♯}}:C^{\bullet }(\mathcal{U},\mathcal{F})⟶C^{\bullet }(\mathcal{V},\mathcal{F})$ defined in the alternating model: for $s\in C^p(\mathcal{U},\mathcal{F})$ and a tuple $(j_0,\dots ,j_p)$ of indices in $J$ one puts $(c^{\mathrm{♯}}s{)}_{j_0\cdots j_p}:=s_{c(j_0)\cdots c(j_p)}{\mid }_{V_{j_0}\cap \cdots \cap V_{j_p}},$ that is, one evaluates the alternating family of $s$ at the tuple of $U$-indices $c(j_0),\dots ,c(j_p)$ and restricts along $V_{j_0}\cap \cdots \cap V_{j_p}\subseteq U_{c(j_0)}\cap \cdots \cap U_{c(j_p)},$ an inclusion holding because $V_j\subseteq U_{c(j)}$ for every $j$ (Sections, restrictions, and global sections of a presheaf).

The displayed families are again alternating cochains, now over $\mathcal{V}$: deleting or permuting entries of $(j_0,\dots ,j_p)$ deletes or permutes the entries of $(c(j_0),\dots ,c(j_p))$, so a repeated index makes the value $0$ by the first alternating condition, while a permutation $\sigma$ of the positions multiplies the value by $\mathrm{sgn}(\sigma )$ by the second condition (Ordered and alternating Čech complexes agree). Hence $c^{\mathrm{♯}}$ is a well-defined homomorphism of abelian groups $C^p(\mathcal{U},\mathcal{F})\to C^p(\mathcal{V},\mathcal{F})$ for every $p$, and it commutes with the Čech differentials, $c^{\mathrm{♯}}\circ {\delta }^{\mathcal{U}}={\delta }^{\mathcal{V}}\circ c^{\mathrm{♯}},$ so that $c^{\mathrm{♯}}$ is a map of cochain complexes. Indeed, deleting the $b$-th entry from the tuple $c(j_0),\dots ,c(j_p)$ gives the tuple $c(j_0),\dots ,\widehat{c(j_b)},\dots ,c(j_p)$ obtained by applying $c$ to the tuple with the $b$-th entry deleted, and the two sides of the identity are the corresponding alternating sums of the sections $s_{c(j_0)\cdots \widehat{c(j_b)}\cdots c(j_p)}$ restricted to $V_{j_0}\cap \cdots \cap V_{j_p}$, which agree by the compatibility of restrictions (Sections, restrictions, and global sections of a presheaf).