Exit paths and constructible stacks

TL;DR

This paper introduces the exit-path 2-category for stratified spaces and proves an equivalence between S-constructible stacks and 2-functors from this category, generalizing fundamental groupoid concepts.

Contribution

It defines the exit-path 2-category for stratified spaces and establishes an equivalence with the 2-category of S-constructible stacks, extending classical topological ideas.

Findings

Defined the exit-path 2-category for stratified spaces

Proved the equivalence between S-constructible stacks and 2-functors from the exit-path 2-category

Generalized the fundamental groupoid concept to a 2-categorical setting

Abstract

For a Whitney stratification S of a space X (or more generally a topological stratification in the sense of Goresky and MacPherson) we introduce the notion of an S-constructible stack of categories on X. The motivating example is the stack of S-constructible perverse sheaves. We introduce a 2-category $EP_{\leq 2}(X,S)$, called the exit-path 2-category, which is a natural stratified version of the fundamental 2-groupoid. Our main result is that the 2-category of S-constructible stacks on X is equivalent to the 2-category of 2-functors from $EP_{\leq 2}(X,S)$ to the 2-category of small categories.