I know that, if $X$ is a locally arcwise connected and locally simply connected topological space, then the restrictions of any locally constant sheaf $\mathcal{F}$ on $X$ corresponding to inclusions $U\subseteq V$ of open subsets which are homotopy equivalences are isomorphisms (this is clear because a locally constant sheaf on $X$ corresponds to a functor $F: \mathbf{ \Pi }_1(X)\rightarrow\mathbf{Set}$).
I was wondering: is it possible to prove the other implication, i.e. whenever $\mathcal{F}$ is a sheaf on $X$ such that $\mathcal{F}(V)\rightarrow\mathcal{F}(U)$ is an isomorphism each time $U\subseteq V$ is an homotopy equivalence, then $\mathcal{F}$ must be locally constant?
Best Answer
This is certainly not true in this generality. For instance, let $X=\mathbb{R}^3\setminus\mathbb{Q}^3$. Then $X$ is locally path connected and locally simply connected, but I'm pretty sure no nontrivial inclusion of open subsets of $X$ is a homotopy equivalence. So, any sheaf at all on $X$ would satisfy your condition.
If you assume $X$ is locally contractible, then it is true. Indeed, the restriction of $\mathcal{F}$ to any contractible open set $U$ is then constant, since we can identify $\mathcal{F}(V)$ with $\mathcal{F}(U)$ for any contractible open $V\subseteq U$ via the restriction map and such $V$ form a basis for the topology.