The homotopy-invariance of constructible sheaves

Abstract

The purpose of this paper is to explain why the functor that sends a stratified topological space $S$ to the $\infty$-category of constructible (hyper)sheaves on $S$ with coefficients in a large class of presentable $\infty$categories is homotopy-invariant. To do this, we first establish a number of results in the unstratified setting, i.e., the setting of locally constant (hyper)sheaves. For example, if $X$ is a locally weakly contractible topological space and $\mathcal{E}$ is a presentable $\infty$-category, then we give a concrete formula for the constant hypersheaf functor $\mathcal{E}\to {\mathrm{Sh}}^{\mathrm{hyp}}(X;\mathcal{E})$. This formula lets us show that the constant hypersheaf functor is a right adjoint, and is fully faithful if $X$ is also weakly contractible. It also lets us prove a general monodromy equivalence and categorical Künneth formula for locally constant hypersheaves.

arXiv comment28 pages. Comments very welcome! v5. Clarified a few things and added some remarks. A slightly shorter version of this paper will appear in Homology, Homotopy and Applications. v4. Completely revised. Two new authors. Much stronger results. v3. Fixed the definition of a "constructible hypersheaf" and clarified a few points. v2. Fixed some typos