Refinement choices induce the same Čech map

Let $c^{\mathrm{♯}},c^{\mathrm{\prime }\mathrm{♯}}$ be the cochain maps of the two refinement functions, so that $V_j\subseteq U_{c(j)}$ and $V_j\subseteq U_{c^{\mathrm{\prime }}(j)}$ for every $j\in J$ [F2] and the displayed formula of [F1] applies to both. For $p\ge 1$ define $h^p:C^p(\mathcal{U},\mathcal{F})\to C^{p-1}(\mathcal{V},\mathcal{F})$ on increasing tuples by the formula of the statement, and set $h^0:=0$. This is well defined: for a tuple $J=(j_0,\dots ,j_{p-1})$ the tuple $M_a:=(c(j_0),\dots ,c(j_a),c^{\mathrm{\prime }}(j_a),\dots ,c^{\mathrm{\prime }}(j_{p-1}))$ has $p+1$ entries, so $s$ may be evaluated there in the alternating model [F3], and $V_{j_0}\cap \cdots \cap V_{j_{p-1}}\subseteq U_{c(j_b)}$ and $\subseteq U_{c^{\mathrm{\prime }}(j_b)}$ for every $b$ by [F2], so the restriction lands in $\mathcal{F}(V_{j_0}\cap \cdots \cap V_{j_{p-1}})$; the map is additive in $s$ because evaluation and restriction are homomorphisms [F8].

Fix a tuple $J=(j_0,\dots ,j_p)$ of pairwise distinct indices and suppress the restrictions to $V_{j_0}\cap \cdots \cap V_{j_p}$, which are compatible [F8]. Put $M_a:=(c(j_0),\dots ,c(j_a),c^{\mathrm{\prime }}(j_a),\dots ,c^{\mathrm{\prime }}(j_p))$ for $0\le a\le p$, a tuple of $p+2$ $U$-indices, and for $0\le a\le p-1$ and $0\le b\le p$ let $N_{a,b}:=M_a(J\setminus b)$ be the mixed tuple formed from the tuple $J$ with the $b$-th entry deleted, so that $N_{a,b}$ has $p+1$ entries. Expanding the definitions, $(h^{p+1}{\delta }^{\mathcal{U}}s{)}_J={\sum }_{a=0}^p{\sum }_{i=0}^{p+1}(-1{)}^{a+i}{s}_{M_a\setminus i}$ and $({\delta }^{\mathcal{V}}{h}^{p}s{)}_J={\sum }_{a=0}^{p-1}{\sum }_{b=0}^p(-1{)}^{a+b}{s}_{N_{a,b}}$, where $M_a\setminus i$ is $M_a$ with the entry at position $i$ removed. Comparing entries gives $N_{a,b}=M_{a+1}\setminus b$ when $b\le a$ and $N_{a,b}=M_a\setminus (b+1)$ when $b>a$: deleting $j_b$ from $J$ and then forming the mixed tuple of split $a$ removes the entry $c(j_b)$, which occupies position $b$ of $M_{a+1}$, in the first case, and removes the entry $c^{\mathrm{\prime }}(j_b)$, which occupies position $b+1$ of $M_a$, in the second.

Match the terms of the two sums. If $i<a$ then the pair $(a-1,i)$ of the second sum is admissible and contributes $N_{a-1,i}=M_a\setminus i$ with coefficient $(-1{)}^{a-1+i}=-(-1{)}^{a+i}$, cancelling the term $(-1{)}^{a+i}{s}_{M_a\setminus i}$ of the first sum; if $i>a+1$ then $a\le p-1$ and the pair $(a,i-1)$ of the second sum is admissible with $i-1>a$, contributing $N_{a,i-1}=M_a\setminus i$ with coefficient $(-1{)}^{a+i-1}=-(-1{)}^{a+i}$ and cancelling the same term. Conversely every term of the second sum is covered: a term $N_{a,b}$ with $b\le a$ is the term $M_{a+1}\setminus b$ of the first sum with $b\le a<a+1$, and a term with $b>a$ is the term $M_a\setminus (b+1)$ of the first sum with $b+1>a+1$. Hence in the sum $(h^{p+1}{\delta }^{\mathcal{U}}s{)}_J+({\delta }^{\mathcal{V}}{h}^{p}s{)}_J$ all terms cancel in pairs except the terms of the first sum with $i=a$ and $i=a+1$, that is $s_{M_a\setminus a}$ and $-s_{M_a\setminus (a+1)}$ for $0\le a\le p$.

It remains to identify the surviving terms. For $0\le a\le p$ the tuple $M_a\setminus a$ deletes the entry $c(j_a)$ at position $a$ of $M_a$ and hence equals $Q_a:=(c(j_0),\dots ,c(j_{a-1}),c^{\mathrm{\prime }}(j_a),\dots ,c^{\mathrm{\prime }}(j_p))$, while $M_a\setminus (a+1)$ deletes the entry $c^{\mathrm{\prime }}(j_a)$ at position $a+1$ of $M_a$ and hence equals $Q_{a+1}=(c(j_0),\dots ,c(j_a),c^{\mathrm{\prime }}(j_{a+1}),\dots ,c^{\mathrm{\prime }}(j_p))$; in particular $Q_0=(c^{\mathrm{\prime }}(j_0),\dots ,c^{\mathrm{\prime }}(j_p))$ and $Q_{p+1}=(c(j_0),\dots ,c(j_p))$. Therefore the surviving terms telescope: $(h^{p+1}{\delta }^{\mathcal{U}}s{)}_J+({\delta }^{\mathcal{V}}{h}^{p}s{)}_J={\sum }_{a=0}^p{s}_{Q_a}-{\sum }_{a=0}^p{s}_{Q_{a+1}}=s_{Q_0}-s_{Q_{p+1}}=s_{c^{\mathrm{\prime }}(j_0)\cdots c^{\mathrm{\prime }}(j_p)}-s_{c(j_0)\cdots c(j_p)}=(c^{\mathrm{\prime }\mathrm{♯}}s{)}_J-(c^{\mathrm{♯}}s{)}_J$, the last equality by the formula of the cochain maps [F1] restricted to $V_{j_0}\cap \cdots \cap V_{j_p}$ [F8].

Since $J$ was an arbitrary tuple of pairwise distinct indices, the identity ${\delta }^{\mathcal{V}}{h}^p+h^{p+1}{\delta }^{\mathcal{U}}=c^{\mathrm{\prime }\mathrm{♯}}-c^{\mathrm{♯}}$ holds as maps $C^p(\mathcal{U},\mathcal{F})\to C^p(\mathcal{V},\mathcal{F})$ for every $p\ge 0$: for $p=0$ the first sum is empty because $h^0=0$, and the second sum consists of the two surviving terms $Q_0,Q_1$ found in [step 2.1] and [step 3.1], which is the same computation. Reading the two cochain complexes as chain complexes by the reindexing convention [F5] and the family $h$ as a degree $+1$ map, this identity is exactly the chain-homotopy relation $f_n-g_n=d_{n+1}^D{s}_n+s_{n-1}{d}_n^C$ of [F4] for $f=c^{\mathrm{\prime }\mathrm{♯}}$, $g=c^{\mathrm{♯}}$ and $s=h$, so $c^{\mathrm{♯}}$ and $c^{\mathrm{\prime }\mathrm{♯}}$ are chain-homotopic. By homotopy invariance [F6] they induce the same map on $H_n$ for every $n$, that is the same map on ${\check{H}}^p(\mathcal{U},\mathcal{F})\to {\check{H}}^p(\mathcal{V},\mathcal{F})$ for every $p$, since ${\check{H}}^p$ is the $p$-th cohomology object [F7] of the Čech complex, computed as a cokernel of cocycles by coboundaries [F9]. ∎