The homotopy-invariance of constructible sheaves

TL;DR

This paper demonstrates that the functor assigning constructible sheaves to stratified spaces remains invariant under homotopy, establishing foundational results in the unstratified setting and deriving key properties like adjointness and monodromy equivalences.

Contribution

It provides the first comprehensive proof of homotopy-invariance for the sheaf functor on stratified spaces, extending classical results to the $ty$-categorical context.

Findings

Established homotopy-invariance of the sheaf functor for stratified spaces.

Derived a concrete formula for the constant hypersheaf functor in locally contractible spaces.

Proved a monodromy equivalence and K"unneth formula for locally constant hypersheaves.

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 K\"unneth formula for locally constant hypersheaves.