A Pair of Quasi-Inverse Functors for an Extension of Perverse Sheaves

TL;DR

This paper introduces a new approach to establish a quasi-inverse functor for the category of perverse sheaves on Thom-Mather spaces, simplifying previous methods by avoiding stack theory and using a different notion called perverse closed set.

Contribution

It proposes a novel concept of perverse closed set to define a quasi-inverse functor, directly addressing the topological case without stack theory.

Findings

Defined a quasi-inverse functor for perverse sheaves

Simplified the equivalence proof without stack theory

Extended the approach to the topological case

Abstract

In their article "Elementary construction of perverse sheaves", R.MacPherson and K. Vilonen show that on a Thom-Mather space X the category PervX of perverse sheaves is equivalent to the category C(F, G, T) whose objects are data of perverse sheaves on the complementary of the closed strata S, a local system on S and some gluing data. To show this equivalence of categories, they define a functor C going from the category PervX to the category C(F, G, T). This definition is based on the notion of perverse link. They do not define a quasi-inverse of this functor. moreover they have to consider first the case where S is contractible and then they extend the equivalence to the topological case using the stack theory. In this paper we propose to consider what we call a perverse closed set which is a bit different from a perverse link in order to define a quasi-inverse to the functor C. Moreover we treat directly the topological case without using stack theory.