Morphisms from the constant sheaf are global sections

For every presheaf of abelian groups $\mathcal{G}$ on $X$ the map ${\mathrm{\Psi }}_{\mathcal{G}}:{\mathrm{Hom}}_{\mathrm{P}\mathrm{A}\mathrm{b}(X)}({\mathbb{Z}}_{\mathrm{p}\mathrm{t}},\mathcal{G})⟶\mathcal{G}(X),\psi \mapsto {\psi }_X(1),$ is a bijection, where $\mathrm{P}\mathrm{A}\mathrm{b}(X)$ denotes presheaves of abelian groups with morphisms given by families of group homomorphisms commuting with restrictions. Injectivity: for a morphism $\psi$ and an open $U$, additivity of ${\psi }_U$ gives ${\psi }_U(n)=n\cdot {\psi }_U(1)$ for all $n\in \mathbb{Z}$, and naturality [F4] with the identity restriction maps of ${\mathbb{Z}}_{\mathrm{p}\mathrm{t}}$ [F1] gives ${\psi }_U(1)={\psi }_X(1){\mathrm{\mid }}_U$ [F6], so ${\psi }_U(n)=n\cdot ({\psi }_X(1){\mathrm{\mid }}_U)$ is determined by ${\psi }_X(1)$. Surjectivity: for $t\in \mathcal{G}(X)$ the formulas ${\psi }_U(n):=n\cdot t{\mathrm{\mid }}_U$ define group homomorphisms that commute with restriction, because $n\cdot t{\mathrm{\mid }}_V=n\cdot (t{\mathrm{\mid }}_U{\mathrm{\mid }}_V)$ for $V\subseteq U$ [F6, F7], and ${\psi }_X(1)=t$. The map is additive because addition of such morphisms is componentwise, so evaluation at $1$ preserves sums.

By [F2] there is a canonical isomorphism of sheaves of sets $\theta :{\mathbb{Z}}_X\to {\underset{‾}{\mathbb{Z}}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$ with each ${\theta }_U$ a bijection, and ${\theta }_U({\eta }_U(a))$ the constant function with value $a$; in particular ${\theta }_X(1_X)$ is the constant function $1$ on $X$. As an isomorphism of sheaves $\theta$ commutes with restrictions, so a section of ${\mathbb{Z}}_X$ over $U$ is a locally constant function $f:U\to \mathbb{Z}$ and for $n\in \mathbb{Z}$ the preimage $f^{-1}(n)\subseteq U$ is open, the sets $f^{-1}(n)$, $n\in \mathbb{Z}$, forming a pairwise disjoint open cover of $U$.

Fix a global section $s\in \mathrm{\Gamma }(X,\mathcal{F})$ and an open $U\subseteq X$ with a locally constant $f:U\to \mathbb{Z}$. For $n\in \mathbb{Z}$ set $u_n:=n\cdot s{\mathrm{\mid }}_{f^{-1}(n)}\in \mathcal{F}(f^{-1}(n))$; since the open sets $f^{-1}(n)$ are pairwise disjoint [step 1.2], the compatibility $u_n{\mathrm{\mid }}_{f^{-1}(n)\cap f^{-1}(m)}=u_m{\mathrm{\mid }}_{f^{-1}(n)\cap f^{-1}(m)}$ holds for all $n,m$, for $n\mathrm{\ne }m$ both sides being sections over the empty open set, and for $n=m$ being the same section, and [F5] provides a unique section ${\psi }_U(f)\in \mathcal{F}(U)$ with ${\psi }_U(f){\mathrm{\mid }}_{f^{-1}(n)}=n\cdot s{\mathrm{\mid }}_{f^{-1}(n)}$ for every $n$. This defines ${\psi }_U$ for every open $U$.

The maps ${\psi }_U$ of [step 2.1] are group homomorphisms and commute with restriction, so they define a morphism of sheaves $\psi :{\mathbb{Z}}_X\to \mathcal{F}$ with ${\psi }_X(1_X)=s$. Additivity: for locally constant $f,g:U\to \mathbb{Z}$ the sections ${\psi }_U(f+g)$ and ${\psi }_U(f)+{\psi }_U(g)$ have the same restriction to each $f^{-1}(n)\cap g^{-1}(m)$, namely $(n+m)\cdot s{\mathrm{\mid }}_{f^{-1}(n)\cap g^{-1}(m)}$, using that restriction $\mathcal{F}(U)\to \mathcal{F}(f^{-1}(n)\cap g^{-1}(m))$ is a group homomorphism [F7] and the defining property of ${\psi }_U$ [step 2.1], so locality [F5] gives ${\psi }_U(f+g)={\psi }_U(f)+{\psi }_U(g)$. Naturality: for $V\subseteq U$ and locally constant $f:U\to \mathbb{Z}$, the restriction $f{\mathrm{\mid }}_V$ is locally constant with $(f{\mathrm{\mid }}_V{)}^{-1}(n)=V\cap f^{-1}(n)$, and both ${\psi }_V(f{\mathrm{\mid }}_V)$ and ${\psi }_U(f){\mathrm{\mid }}_V$ restrict to $n\cdot s{\mathrm{\mid }}_{V\cap f^{-1}(n)}=n\cdot (s{\mathrm{\mid }}_{f^{-1}(n)}){\mathrm{\mid }}_{V\cap f^{-1}(n)}$ [F6, F7], so locality gives ${\psi }_V(f{\mathrm{\mid }}_V)={\psi }_U(f){\mathrm{\mid }}_V$ [F4, F5]. Finally ${\psi }_X(1_X)=s$ because ${\theta }_X(1_X)$ is the constant function $1$ [step 1.2], whose level set in degree $1$ is all of $X$ and whose level sets in degrees $n\mathrm{\ne }1$ are empty.

By [step 3.1] every $s\in \mathrm{\Gamma }(X,\mathcal{F})$ equals ${\psi }_X(1_X)={\mathrm{\Phi }}_{\mathcal{F}}(\psi )$ for some morphism $\psi :{\mathbb{Z}}_X\to \mathcal{F}$; hence ${\mathrm{\Phi }}_{\mathcal{F}}$ is surjective.

Let $\phi :{\mathbb{Z}}_X\to \mathcal{F}$ be a morphism and put $s:={\phi }_X(1_X)$. For every open $U$ and every $n\in \mathbb{Z}$ the section ${\eta }_U(n)$ is the constant function $n$ on $U$ [step 1.2, F3], so naturality of $\phi$ [F4] and additivity of ${\phi }_U$ give $(\phi \circ \eta {)}_U(n)={\phi }_U({\eta }_U(n))=n\cdot {\phi }_U({\eta }_U(1))=n\cdot s{\mathrm{\mid }}_U$, because ${\eta }_U(1)=1_X{\mathrm{\mid }}_U$ [F6]; the same value $n\cdot s{\mathrm{\mid }}_U$ is obtained from the morphism $\psi$ of [step 3.1] by its defining property [step 2.1]. Hence $\phi \circ \eta =\psi \circ \eta$, and the uniqueness part of [F10] forces $\phi =\psi$. Thus ${\mathrm{\Phi }}_{\mathcal{F}}$ is injective with inverse $s\mapsto \psi$. Moreover the computation ${\phi }_U(f){\mathrm{\mid }}_{f^{-1}(n)}=n\cdot s{\mathrm{\mid }}_{f^{-1}(n)}$ displayed in clause 1 follows from naturality and additivity of ${\phi }_U$ [F8] applied to $f{\mathrm{\mid }}_{f^{-1}(n)}$, the constant function $n$ over $f^{-1}(n)$ [step 1.2], and agrees with the description of $\psi$ in [step 2.1].

The map ${\mathrm{\Phi }}_{\mathcal{F}}$ is additive: for morphisms $\phi ,{\phi }^{\mathrm{\prime }}:{\mathbb{Z}}_X\to \mathcal{F}$ the identity $(\phi +{\phi }^{\mathrm{\prime }}{)}_X={\phi }_X+{\phi }_X^{\mathrm{\prime }}$ of componentwise addition [F8] gives ${\mathrm{\Phi }}_{\mathcal{F}}(\phi +{\phi }^{\mathrm{\prime }})={\mathrm{\Phi }}_{\mathcal{F}}(\phi )+{\mathrm{\Phi }}_{\mathcal{F}}({\phi }^{\mathrm{\prime }})$. Together with [step 4.1] and [step 4.2] this makes ${\mathrm{\Phi }}_{\mathcal{F}}$ an isomorphism of abelian groups. Naturality: for a morphism $\psi :\mathcal{F}\to \mathcal{G}$ of abelian sheaves and $\phi :{\mathbb{Z}}_X\to \mathcal{F}$ one has ${\mathrm{\Phi }}_{\mathcal{G}}(\psi \circ \phi )=(\psi \circ \phi {)}_X(1_X)={\psi }_X({\phi }_X(1_X))=\mathrm{\Gamma }(X,\psi )({\mathrm{\Phi }}_{\mathcal{F}}(\phi ))$ by componentwise composition of families [F8].

Let $J^{\bullet }$ be a cochain complex of abelian sheaves and read ${\mathbb{Z}}_X$ as the stalk complex concentrated in degree $0$ [F11]. In the cochain Hom complex the $n$-th entry is ${\prod }_m\mathrm{Hom}({\mathbb{Z}}_X^m,J^{m+n})$ [F12], and since ${\mathbb{Z}}_X^m=0$ for $m\mathrm{\ne }0$ [F11] this entry is $\mathrm{Hom}({\mathbb{Z}}_X,J^n)$, with differential $du=d_{J^n}u-(-1{)}^{n}u{d}_{{\mathbb{Z}}_X}$ and $d_{{\mathbb{Z}}_X}=0$ [F11, F12], that is, $du=d_{J^n}\circ u$. Transporting along the bijections ${\mathrm{\Phi }}_{J^n}$ of [step 5.1] sends $u$ to ${\mathrm{\Phi }}_{J^n}(u)=u_X(1_X)$, and naturality of $\mathrm{\Phi }$ [step 5.1] turns composition with $d_{J^n}$ into the map $\mathrm{\Gamma }(X,d^n)$, so the levelwise maps define a cochain map ${\underset{}{\mathrm{Hom}}}^{\bullet }({\mathbb{Z}}_X,J^{\bullet })\to \mathrm{\Gamma }(X,J^{\bullet })$ [F13] that is bijective in every degree, hence an isomorphism of cochain complexes, natural in $J^{\bullet }$ because each ${\mathrm{\Phi }}_{J^n}$ is.

Clause 1 is [step 4.1] together with [step 4.2], the description of the inverse being the one constructed in [step 2.1] and verified in [step 3.1] and [step 4.2]; clause 2 is [step 5.1]; clause 3 is [step 6.1]. No choice principle is used anywhere. ∎