Stacks similar to the stack of perverse sheaves

Treumann, David

doi:10.1090/S0002-9947-2010-04958-X

Stacks similar to the stack of perverse sheaves
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Trans. Amer. Math. Soc. 362 (2010), 5395-5409 Request permission

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”.

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