I would like to know if sheaf theory can be applied to a particular class of questions in topology.
The Schwarz genus (also known as sectional category) of a continuous map $p\colon\thinspace E\to B$ is the smallest integer $k$ such that $B$ can be covered by open subsets $U_1,\ldots ,U_k$ over each of which $p$ admits a local section, ie a continuous map $s_i\colon\thinspace U_i\to E$ such that $p\circ s_i$ equals the inclusion $U_i\hookrightarrow B$. If no such cover exists (eg if $p$ is not surjective) we set the genus to be $\infty$.
Several important numerical invariants in topology are special cases of this genus. For instance, the Lusternik-Schnirelmann category $\mathrm{cat}(X)$ of a space $X$, defined to be the smallest $k$ such that $X$ can be covered by open subsets $U_1,\ldots ,U_k$ such that each inclusion $U_i\hookrightarrow X$ is null-homotopic, is easily seen to be the genus of the Serre fibration $PX\to X$ of based paths on $X$. More recently, Farber has defined the topological complexity $\mathrm{TC}(X)$ of a space $X$ to be the genus of the fibration $X^I\to X\times X$ which takes a free path in $X$ to its pair of initial and final points, and this is relevant to the motion planning problem in robotics.
On the other hand, one of the most natural examples of a sheaf (at least for a topologist) is the sheaf of sections of a continuous map $p\colon\thinspace E\to B$. This is the sheaf $\Gamma(p)$ on $B$ whose sections over an open set $U\subseteq B$ are the set $\Gamma(p)(U)$ of local sections $s\colon\thinspace U\to E$ of $p$, as defined above.
This leads me to the following (perhaps naive) question.
Definition. Let $\mathcal{F}$ be a sheaf of sets over $X$. Define the Schwarz genus of $\mathcal{F}$ to be the least $k$
such that $X$ has a cover by open subsets $U_1,\ldots, U_k$ such that each $\mathcal{F}(U_i)\neq\emptyset$.Do there exist techniques in sheaf theory to approximate (bound from above or below) the Schwarz genus of $\mathcal{F}$? Has this invariant been considered before?
I suspect the answer is no, but I would like to hear this from a sheafy person (of which there seem to plenty on MO). Also I would be interested to hear of any extra conditions
you would impose to make the question more interesting or tractable.
Edit: As Ben noted below, we can apply the free abelian group functor to $\mathcal{F}$ to get a sheaf $\mathcal{G}$ of abelian groups over $X$. Define the Schwarz genus of $\mathcal{G}$ to be the least $k$ such that $X$ has a cover by open subsets $U_1,\ldots, U_k$ such that each $\mathcal{G}(U_i)\neq 0$. Can we obtain bounds on this genus using the Cech spectral sequence, or something similar?
