Sheaf cohomology as right derived global sections

Assume the Axiom of Choice. Fix a topological space $X$, let $\mathrm{\Gamma }(X,-):\mathrm{A}\mathrm{b}(X)\to \mathbf{A}\mathbf{b}$ be the additive left exact global-sections functor (Global sections of an abelian sheaf), and let $I$ be the supplied functorial injective resolution datum on $\mathrm{A}\mathrm{b}(X)$ of Enough injective abelian sheaves: it assigns to every abelian sheaf $\mathcal{F}$ one specific injective resolution $0\to \mathcal{F}\to I^{\bullet }(\mathcal{F})$ (Right derived objects relative to supplied injective resolution data).

For every $q\ge 0$ and every abelian sheaf $\mathcal{F}$ on $X$ define the $q$-th sheaf cohomology group of $\mathcal{F}$ to be the right derived object of $\mathrm{\Gamma }(X,-)$ relative to $I$: $H^q(X,\mathcal{F}):=R_I^q\mathrm{\Gamma }(X,\mathcal{F})=H^q(\mathrm{\Gamma }(X,I^{\bullet }(\mathcal{F}{)}_{\mathrm{d}\mathrm{e}\mathrm{l}})),$ the cohomology object of the complex $\mathrm{\Gamma }(X,I^0(\mathcal{F}))\to \mathrm{\Gamma }(X,I^1(\mathcal{F}))\to \cdots$ of abelian groups obtained from the deleted resolution (Deleted resolutions, Cohomology object of a cochain complex). We also set $H^q(X,\mathcal{F}):=0$ for $q<0$.

Because $I$ is a fixed supplied datum, the group $H^q(X,\mathcal{F})$ is a specific group for each pair $(X,\mathcal{F})$, and for a morphism $\phi :\mathcal{F}\to \mathcal{G}$ of abelian sheaves we write $H^q(X,\phi ):=R_I^q\mathrm{\Gamma }(X,\phi )$ for the induced map; it is the map on cohomology induced by $\mathrm{\Gamma }(X,-)$ applied to the cochain maps supplied with the datum, and it is additive in $\phi$.

If $J$ is another supplied injective resolution datum on $\mathrm{A}\mathrm{b}(X)$ then $R_I^q\mathrm{\Gamma }(X,-)$ and $R_J^q\mathrm{\Gamma }(X,-)$ are naturally isomorphic, so the $H^q(X,\mathcal{F})$ computed from $J$ agree with the ones above up to a canonical natural isomorphism; the Axiom of Dependent Choice needed for that comparison follows from AC (Two supplied injective resolution data define naturally isomorphic right derived functors, AC implies DC implies countable choice). The notation $H^q(X,\mathcal{F})$ therefore does not depend on the datum used, up to this canonical isomorphism, and we call $q$ the cohomological degree.