Cohomology of the empty space and the empty cover

The only open subset of $X=\mathrm{\varnothing }$ is $\mathrm{\varnothing }$ itself, since open sets are subsets of $X$; in particular the family $\mathcal{U}:=\mathrm{\varnothing }\subseteq \mathcal{T}$ consisting of no open subset is a family of open subsets of $X$, and its union, $\bigcup \mathcal{U}=\{x\in X:x\in U\text{ for some }U\in \mathcal{U}\}$, is empty because there is no member to witness membership, that is $\bigcup \mathcal{U}=\mathrm{\varnothing }=X$. By the definition of an open cover [F6] the empty family is therefore an open cover of the empty space, and reading it as an indexed family $\mathcal{U}=(U_i{)}_{i\in \mathrm{\varnothing }}$ with index set $I=\mathrm{\varnothing }$ and no members exhibits it as a cover in the indexed form used by the Čech construction [F2]; the empty space thus carries the cover with no members.

For every $p\ge 0$ the index set $I=\mathrm{\varnothing }$ has no increasing $(p+1)$-tuple $i_0<\cdots <i_p$, since it has no elements at all, so by the empty-product convention of [F2] the group of ordered $p$-cochains is $C^p(\mathcal{U},\mathcal{F})=0$ for every $p\ge 0$; for $p<0$ one has $C^p(\mathcal{U},\mathcal{F})=0$ by the same definition. In particular $C^0=0$ and $C^1=0$, and more generally ${\delta }^p:C^p\to C^{p+1}$ is a homomorphism between zero groups, hence the zero homomorphism. By [F3] the cover's cohomology is ${\check{H}}^p(\mathcal{U},\mathcal{F})=\ker {\delta }^p\mathrm{/}\mathrm{im}{\delta }^{p-1}$, the kernel of the zero map out of the zero group is the zero group, and the image of ${\delta }^{p-1}$ is the zero subgroup of $C^p$ for every $p\ge 0$, the case $p=0$ being the case of the zero map ${\delta }^{-1}=0$; hence ${\check{H}}^p(\mathcal{U},\mathcal{F})=0\mathrm{/}0=0$ for every $p\ge 0$ and, by the convention of [F3], also for $p<0$. Thus the empty cover has zero cochain groups and zero cohomology in every degree, for every abelian sheaf $\mathcal{F}$ on $X$.

Let $\mathcal{F}$ be a sheaf of abelian groups on $X$. The underlying sheaf of sets has $\mathcal{F}(\mathrm{\varnothing })$ a singleton by [F1]; a group whose underlying set is a singleton is the trivial group, so $\mathcal{F}(\mathrm{\varnothing })=0$. Since $\mathrm{\varnothing }$ is the only open subset of $X$ [step 1.1], every section group $\mathcal{F}(U)$ with $U$ open in $X$ equals $\mathcal{F}(\mathrm{\varnothing })=0$.

By the definition of the global-sections functor [F5] one has $\mathrm{\Gamma }(X,\mathcal{G})=\mathcal{G}(X)=\mathcal{G}(\mathrm{\varnothing })$ for every abelian sheaf $\mathcal{G}$ on $X$, and $\mathrm{\Gamma }(X,\phi )={\phi }_X$ for every morphism $\phi :\mathcal{G}\to {\mathcal{G}}^{\mathrm{\prime }}$. By [step 2.2] applied to $\mathcal{G}$ the group $\mathcal{G}(\mathrm{\varnothing })$ is zero, so $\mathrm{\Gamma }(X,\mathcal{G})=0$ for every object $\mathcal{G}$ of $\mathrm{A}\mathrm{b}(X)$, while $\mathrm{\Gamma }(X,\phi )$ is the only map between the zero groups $\mathcal{G}(\mathrm{\varnothing })\to {\mathcal{G}}^{\mathrm{\prime }}(\mathrm{\varnothing })$, namely the zero map; hence $\mathrm{\Gamma }(X,-):\mathrm{A}\mathrm{b}(X)\to \mathbf{A}\mathbf{b}$ is the zero functor, constant with value the zero group.

Fix the supplied functorial injective resolution datum $I$ on $\mathrm{A}\mathrm{b}(X)$, so that every abelian sheaf $\mathcal{F}$ on $X$ is equipped with a specific injective resolution $0\to \mathcal{F}\to I^{\bullet }(\mathcal{F})$ and $H^q(X,\mathcal{F})=R_I^q\mathrm{\Gamma }(X,\mathcal{F})=H^q(\mathrm{\Gamma }(X,I^{\bullet }(\mathcal{F}{)}_{\mathrm{d}\mathrm{e}\mathrm{l}}))$ by [F4]. Applying the zero functor of [step 3.1] term by term, every group of the deleted complex $\mathrm{\Gamma }(X,I^{\bullet }(\mathcal{F}{)}_{\mathrm{d}\mathrm{e}\mathrm{l}})$ is the zero group and every differential of it is the zero map, so in the notation of [F7] both $Z^q(C)=\ker (d^q)$ and $B^q(C)=\mathrm{im}(d^{q-1})$ are the zero subgroup of the zero group $C^q=0$; hence $H^q(X,\mathcal{F})=\mathrm{coker}(B^q\to Z^q)=0\mathrm{/}0=0$ for every $q\ge 0$. By the convention recorded in [F4] one also has $H^q(X,\mathcal{F})=0$ for $q<0$, so every abelian sheaf on the empty space is acyclic for $\mathrm{\Gamma }(X,-)$ in all degrees.

Combining the two computations: [step 2.1] shows that the empty cover of $X=\mathrm{\varnothing }$ has $C^p(\mathcal{U},\mathcal{F})=0$ for every $p\ge 0$ and ${\check{H}}^p(\mathcal{U},\mathcal{F})=0$ for every $p$, and [step 4.1] shows that $H^q(X,\mathcal{F})=0$ for every abelian sheaf $\mathcal{F}$ on $X$ and every $q\ge 0$; the two statements together are the assertion of the statement, the equality $\mathcal{F}(\mathrm{\varnothing })=0$ of [step 2.2] being the reason why the global-sections functor is the zero functor. The Axiom of Choice is assumed and is used exactly once: the definition [F4] of $H^q(X,\mathcal{F})$ as a right derived object relative to the supplied injective resolution datum, whose independence of the chosen datum rests on the Axiom of Dependent Choice that follows from AC [F8]; the empty cover and the vanishing of the section groups in [step 1.1], [step 2.1], [step 2.2], [step 3.1] and [step 4.1] use no choice principle at all, the only products occurring there being indexed by the empty set, and a product over the empty set of groups is the one-element group by definition and not by [F9]. ∎