2021/02/24 by Damien Lejay, Lejay, Damien · 1 citation
Computer Science · Mathematics · #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2102.12325
openalex publication_date 2021/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The goal of this article is to extend a theorem of Lurie ShA (X) = Fun(ExitA (X), S) representing constructible sheaves with values in S , the ∞ -category of spaces, on a stratified space X with poset of strata A , as functors from the exit paths ∞ -category ExitA (X) to 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≤ 0 ⊂ A≤ 1 ⊂ ⋯ ⊂ A by posets satisfying the ascending chain condition, HypA (X) = Fun(ExitA(X), S) the ∞ -category of A -constructible hypersheaves on X is represented by functors from the exit paths ∞ -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.