Stacks similar to the stack of perverse sheaves

TL;DR

This paper introduces a new class of stacks called 'stacks of type P' on topological spaces, generalizing the structure of perverse sheaves and establishing foundational properties and equivalences for these stacks.

Contribution

It defines stacks of type P, shows they share key structural features with perverse sheaves, and develops a tilting formalism for these stacks.

Findings

Stacks of type P are locally equivalent to MacPherson-Vilonen constructions.

Under connectedness conditions, their global sections correspond to modules over finite-dimensional algebras.

The paper develops a tilting formalism for stacks of type P.

Abstract

We introduce, on a topological space X, a class of stacks of abelian categories we call "stacks of type P." This class of stacks includes the stack of perverse sheaves (of any perversity, constructible with respect to a fixed stratification), and is singled out by fairly innocuous axioms. We show that some basic structure theory for perverse sheaves holds for a general stack of type P: such a stack is locally equivalent to a MacPherson-Vilonen construction, and under certain connectedness conditions its category of global objects is equivalent to the category of modules over a finite-dimensional algebra. To prove these results we develop a rudimentary tilting formalism for stacks of type P -- another sense in which these stacks are "similar to stacks of perverse sheaves."