Cohomological dimension relative to a sheaf class

Assume the Axiom of Choice (The Axiom of Choice). Let $X$ be a topological space, let $\mathcal{C}$ be a class of sheaves of abelian groups on $X$ (that is, a subclass of the objects of $\mathrm{A}\mathrm{b}(X)$, for example the class of all abelian sheaves, or a class of sheaves of modules over a fixed sheaf of rings), and let $H^q(X,-)$ be sheaf cohomology relative to the supplied functorial injective resolution datum (Sheaf cohomology as right derived global sections). The cohomological dimension of $X$ relative to $\mathcal{C}$ is ${\mathrm{cd}}_{\mathcal{C}}(X):=\inf \{d\in \mathbb{Z}:H^q(X,\mathcal{F})=0\text{ for every }\mathcal{F}\in \mathcal{C}\text{ and every }q>d\},$ the infimum of the empty set of integers being $+\mathrm{\infty }$. Thus:

1. ${\mathrm{cd}}_{\mathcal{C}}(X)\le d$ if and only if $H^q(X,\mathcal{F})=0$ for every $\mathcal{F}\in \mathcal{C}$ and every $q>d$; the value $+\mathrm{\infty }$ means that for every integer $d$ the class $\mathcal{C}$ contains a sheaf with $H^q(X,\mathcal{F})\mathrm{\ne }0$ for some $q>d$;
2. if $H^0(X,\mathcal{F})\mathrm{\ne }0$ for some $\mathcal{F}\in \mathcal{C}$, then ${\mathrm{cd}}_{\mathcal{C}}(X)\in {\mathbb{Z}}_{\ge 0}\cup \{+\mathrm{\infty }\}$. If this value is finite, the infimum is attained and ${\mathrm{cd}}_{\mathcal{C}}(X)=\min \{d\ge 0:H^q(X,\mathcal{F})=0\text{ for all }\mathcal{F}\in \mathcal{C},q>d\}$, the least nonnegative integer with the vanishing property. If there is no finite uniform bound, then ${\mathrm{cd}}_{\mathcal{C}}(X)=+\mathrm{\infty }$ and the infimum is not attained;
3. ${\mathrm{cd}}_{\mathcal{C}}(X)=-\mathrm{\infty }$ exactly when $H^q(X,\mathcal{F})=0$ for every $\mathcal{F}\in \mathcal{C}$ and every $q\ge 0$; in particular ${\mathrm{cd}}_{\mathcal{C}}(\mathrm{\varnothing })=-\mathrm{\infty }$ for every class $\mathcal{C}$ on the empty space, since $\mathcal{F}(\mathrm{\varnothing })$ is the one-element group and $\mathrm{\Gamma }(\mathrm{\varnothing },-)$ is exact (A set-valued sheaf has a unique section over the empty open set, An exact functor has vanishing positive derived functors);
4. ${\mathrm{cd}}_{\mathcal{C}}(X)\le {\mathrm{cd}}_{{\mathcal{C}}^{\mathrm{\prime }}}(X)$ whenever $\mathcal{C}\subseteq {\mathcal{C}}^{\mathrm{\prime }}$; when $\mathcal{C}$ is the class of all abelian sheaves on $X$ one writes $\mathrm{cd}(X)$ for ${\mathrm{cd}}_{\mathcal{C}}(X)$ and calls it the cohomological dimension of $X$;
5. the definition is independent of the supplied resolution datum up to the canonical isomorphisms of Sheaf cohomology as right derived global sections, so ${\mathrm{cd}}_{\mathcal{C}}(X)$ depends on $X$ and $\mathcal{C}$ only.

In degree zero the vanishing hypothesis in item 2 is satisfied by any class containing a sheaf with a nonzero global section, by Degree-zero sheaf cohomology is global sections.