[Paper Review] Stacks similar to the stack of perverse sheaves

David Treumann|ArXiv.org|Jan 19, 2008

Algebraic structures and combinatorial models9 references4 citations

TL;DR

This paper introduces 'stacks of type P'—a class of stacks of abelian categories on stratified spaces that generalize the stack of perverse sheaves. It proves that such stacks are locally equivalent to MacPherson-Vilonen constructions and, under 2-connectedness conditions, their global sections form module categories over finite-dimensional algebras, via a developed tilting formalism.

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."

Motivation & Objective

• To define and characterize a broad class of stacks—'stacks of type P'—that generalize the stack of perverse sheaves.
• To show that these stacks share key structural properties with perverse sheaves, such as local equivalence to MacPherson-Vilonen constructions.
• To establish a finiteness result: under 2-connectedness of strata, global sections of a stack of type P are equivalent to modules over a finite-dimensional algebra.
• To develop a tilting formalism for stacks of type P, mirroring that of perverse sheaves, to support structural analysis.

Proposed method

• Define stacks of type P via four axioms: constructibility with respect to a stratification, local finiteness of Hom and Ext groups, uniqueness of simple objects per stratum, and Serre quotient structure for closed unions of strata.
• Use induction on the number of strata, leveraging regular neighborhoods of closed strata to reduce to local models.
• Apply the MacPherson-Vilonen construction to stalks of the stack, showing that each stalk is equivalent to a category $ C(F,G;T) $ of triples involving functors and natural transformations.
• Construct a retraction functor from the global category to the category of the closed stratum, inducing an equivalence to a MacPherson-Vilonen construction.
• Use weak pullback squares to relate global categories to local data, ensuring compatibility across open covers.
• Leverage the tilting formalism to show that the global category has enough projectives, enabling the conclusion that it is equivalent to modules over a finite-dimensional algebra.

Experimental results

Research questions

• RQ1 Can the structure of the stack of perverse sheaves be generalized to a broader class of stacks with similar categorical properties?
• RQ2 Is every stack of type P locally equivalent to a MacPherson-Vilonen construction?
• RQ3 Under what topological conditions on the stratification does the category of global sections of a stack of type P become equivalent to the category of modules over a finite-dimensional algebra?
• RQ4 How can a tilting formalism be developed for stacks of type P to mirror that of perverse sheaves?

Key findings

• The stack of $\mathcal{S}$-constructible $p$-perverse sheaves on a stratified space $X$ is a stack of type P, generalizing the classical case.
• Every stalk of a stack of type P is equivalent to a MacPherson-Vilonen construction $C(F,G;T)$, where $F$ and $G$ are functors from the stalk category to vector spaces.
• When all strata are 2-connected, the category of global sections $\mathcal{C}(X)$ is equivalent to the category of finite-dimensional modules over a finite-dimensional $\mathbb{F}$-algebra.
• The tilting formalism developed for stacks of type P enables the construction of retractions and equivalences to MacPherson-Vilonen categories.
• The global category $\mathcal{C}(X)$ has enough projectives, a key property for module category equivalence, as established via results from [11].
• The weak pullback structure of the stack ensures compatibility of local data, allowing the global category to be reconstructed from local MacPherson-Vilonen data.