Exodromy beyond conicality
Abstract
We show that compact subanalytic stratified spaces and algebraic stratifications of real varieties have finite exitpath $\infty$categories, refining classical theorems of LefschetzWhitehead, Lojasiewicz, and Hironaka on the finiteness of the underlying homotopy types of these spaces. These stratifications are typically not conical; hence we cannot rely on the currently available exodromy equivalence between constructible sheaves on a stratified space, which requires conicality as a fundamental hypothesis. Building on ideas of Clausen and Orsnes Jansen, we study the class of exodromic stratified spaces, for which the conclusion of the exodromy theorem holds. We prove two new fundamental properties of this class of stratified spaces: coarsenings of exodromic stratifications are exodromic, and every morphism between exodromic stratified spaces induces a functor between the associated exit path $\infty$categories. As a consequence, we produce many new examples of exodromic stratified spaces, including: coarsenings of conical stratifications, locally finite subanalytic stratifications of real analytic spaces, and algebraic stratifications of real varieties. Our proofs are at the generality of stratified $\infty$topoi, hence apply to even more general situations such as stratified topological stacks. Finally, we use the previously mentioned finiteness results to construct derived moduli stacks of constructible and perverse sheaves.
 Publication:

arXiv eprints
 Pub Date:
 January 2024
 DOI:
 10.48550/arXiv.2401.12825
 arXiv:
 arXiv:2401.12825
 Bibcode:
 2024arXiv240112825H
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Algebraic Geometry;
 Mathematics  Category Theory
 EPrint:
 Comments very welcome. 73 pages