Constructible hypersheaves via exit paths

Abstract

The goal of this article is to extend a theorem of Lurie \[ \mathsf{Sh}_A (X) = \mathsf{Fun}(\mathsf{Exit}_A (X), \mathsf{S}) \] representing constructible sheaves with values in $ \mathsf{S} $, the $ \infty $-category of spaces, on a stratified space $ X $ with poset of strata $ A $, as functors from the exit paths $ \infty $-category $ \mathsf{Exit}_A (X) $ to $ \mathsf{S} $. Lurie's representation theorem works provided $ A $ satisfy the ascending chain condition. This typically rules out infinite dimensional examples of stratified space. Building on it and with the help of a stratified homotopy invariance theorem from Haine, we show that when $ X $ is a nice enough $ A $-stratified space and when $ A $ is itself stratified $ A_{\leq 0} \subset A_{\leq 1} \subset \cdots \subset A $ by posets satisfying the ascending chain condition, \[ \mathsf{Hyp}_A (X) = \mathsf{Fun}(\mathsf{Exit}_A(X), \mathsf{S}) \] the $ \infty $-category of $ A $-constructible hypersheaves on $ X $ is represented by functors from the exit paths $ \infty $-category of $ X $. There are two types of nice stratified spaces on which this extended representation theorem applies: conically stratified spaces and spaces that are sequential colimits of conically stratified spaces. Examples of application include the metric and the topological exponentials of a Fr\'echet manifold, locally countable simplicial complexes and more generally, locally countable cylindrically normal CW-complexes.