The homotopy-invariance of constructible sheaves

In this paper we show that the functor sending 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 for 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 $\mathcal{E} \to \mathrm{Sh}^\mathrm{hyp} (X; \mathcal{E})$, implying 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.

Keywords

locally constant sheaf, constructible sheaf, hypersheaf, homotopy-invariance

2010 Mathematics Subject Classification

32S60, 55N05, 55N30, 55P55

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Received 21 February 2022

Received revised 13 August 2022

Accepted 13 August 2022

Published 4 October 2023