Verdier duality on conically smooth stratified spaces

TL;DR

This paper establishes a Verdier duality for constructible sheaves on conically smooth stratified spaces, extending classical duality results to a broader geometric setting using advanced $alculus and singularity resolution techniques.

Contribution

It proves a duality theorem for constructible sheaves in the setting of conically smooth stratified spaces, employing the geometry of these spaces and resolution of singularities.

Findings

Verdier duality restricts to an equivalence between constructible sheaves and cosheaves.

Computed the exit path $alculus$-category of a compact stratified space.

Extended duality results to conically smooth stratified spaces using singularity resolution.

Abstract

In this paper we prove a duality for constructible sheaves on conically smooth stratified spaces. Here we consider sheaves with values in a stable and bicomplete $\infty$-category equipped with a closed symmetric monoidal structure, and in this setting constructible means locally constant along strata and with dualizable stalks. The crucial point where we need to employ the geometry of conically smooth structures is in showing that Lurie's version of Verdier duality restricts to an equivalence between constructible sheaves and cosheaves: this requires a computation of the exit path $\infty$-category of a compact stratified space, that we obtain via resolution of singularities.