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 $A_{\le 0}\subset A_{\le 1}\subset \cdots \subset A$ 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 Fréchet manifold, locally countable simplicial complexes and more generally, locally countable cylindrically normal CW-complexes.

30 pages