Exit paths and constructible stacks

Abstract

For a Whitney stratification S of a space X (or, more generally, a topological stratification in the sense of Goresky and MacPherson) we introduce the notion of an S-constructible stack of categories on X. The motivating example is the stack of S-constructible perverse sheaves. We introduce a 2-category EP 2 (X, S), called the exit-path 2-category, which is a natural stratified version of the fundamental 2-groupoid. Our main result is that the 2-category of S-constructible stacks on X is equivalent to the 2-category of 2-functors 2Funct(EP 2 (X, S), Cat) from the exit-path 2-category to the 2-category of small categories.

Stacks similar to the stack of perverse sheaves

“…(More plausibly, to a more high-tech setting such as stacks of dg categories or stable ∞-categories.) This paper and [13] constitute, in part, an effort to start developing a structure theory for such stacks. See for instance Remark 3.2.…”

“…Here L is a compact topologically stratified space of lower dimension, and CL denotes the open cone on L along with its induced decomposition. We refer to [9] and [13] for more detailed definitions.…”

“…The most basic of these is that a stack of type P should be constructible with respect to a stratification of X. (The notion of a constructible stack, directly analogous to the notion of a constructible sheaf, is introduced in [13].) We investigate the extent to which these stacks behave "similarly to stacks of perverse sheaves".…”

Dualities between cellular sheaves and cosheaves

“…A second reason is that if a sheaf is locally constant when restricted to each cell, then one can pushforward and pullback this sheaf along the map q : |X| → X described in Remark 4.6 to obtain an isomorphic sheaf. A third reason is that (co)sheaves that are constructible with respect to a cell structure are equivalent to cellular (co)sheaves [4,8,11,20].…”

The non-equivariant coherent-constructible correspondence and a conjecture of King

“…We make use of the convenient recent reference [Ba], but remark that this circle of ideas goes back to McPherson and others: for more information see for instance [Ka], and the discussion in the first Section of [Tr1].…”