Constructible hypersheaves via exit paths

TL;DR

This paper extends Lurie's theorem to represent constructible hypersheaves on certain stratified spaces as functors from exit path categories, including infinite-dimensional cases, using stratified homotopy invariance.

Contribution

It generalizes the representation of constructible sheaves to a broader class of stratified spaces, including those with infinite dimensions, by leveraging stratified homotopy invariance.

Findings

Representation theorem applies to conically stratified spaces.

Applicable to spaces as colimits of conically stratified spaces.

Includes examples like metric and topological exponentials of Fréchet manifolds.

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.