The homotopy-invariance of constructible sheaves

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.