Exit paths and constructible stacks

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.

💡 Research Summary

The paper develops a higher‑categorical framework for constructible sheaf‑like objects on a stratified space. Let (X) be a topological space equipped with a Whitney (or more generally a Goresky‑MacPherson) stratification (S). The authors introduce the notion of an (S)-constructible stack of categories: a stack on (X) whose stalk at each point is a small category, and whose restriction functors respect the stratification in the sense that they are defined only along “exit paths” that move from higher‑dimensional strata to lower‑dimensional ones. This generalises the classical constructible perverse sheaf, which can be viewed as a stack of abelian categories with additional t‑structure data.

The central new object is the exit‑path 2‑category (EP_{\le 2}(X,S)). Its objects are the points of (X); a 1‑morphism from (x) to (y) is an exit path—a continuous map (\gamma: