Stacks similar to the stack of perverse sheaves

Abstract

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".

Exit paths and constructible stacks

“…The stack property together with the MacPherson-Vilonen construction give an inductive strategy for computing categories of perverse sheaves. One of the motivations for the theory in this paper is to analyze this strategy systematically; see [23].…”

“…This composition is strictly associative, and it has strict units, so this data defines a 2-category 2Funct(C, D). We define 2-natural transformations here; more details may be found in[11]:Definition B 23…”

Faisceaux pervers sur les variétés toriques lisses

“…Ceci permettrait, par exemple, de décrire les faisceaux pervers sur un espace qui est localement un des espaces cités. Dans [9] et [8], D. Treumann utilise se genre de méthodes pour généraliser le procédé défini dans [6].…”

Stacks on stratified space

“…As an application , in [19], D. Treumann has used his description of the 2-category of constructible stacks and a description of the category of perverse sheaves given by MacPherson and Vilonen in [13] to characterize the stack of perverse sheaves and in the case of Thom-Mather spaces he has showed that if the stratum are 2-connected the category of perverse sheaves is equivalent to the category of finite-dimensional modules over a finite-dimensional algebra. As he has used a non explicit local description he does not obtain an explicit description.…”