  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.
