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Exodromy beyond conicality

Peter J. Haine, Mauro Porta, Jean-Baptiste Teyssier

Abstract

We show that compact subanalytic stratified spaces and algebraic stratifications of real varieties have finite exit-path $\infty$-categories, refining classical theorems of Lefschetz-Whitehead, 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.

Exodromy beyond conicality

Abstract

We show that compact subanalytic stratified spaces and algebraic stratifications of real varieties have finite exit-path -categories, refining classical theorems of Lefschetz-Whitehead, 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 -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 -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.
Paper Structure (43 sections, 104 theorems, 245 equations, 1 figure)

This paper contains 43 sections, 104 theorems, 245 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Exodromic stratified spaces
  4. The stability theorem
  5. Applications of the stability theorem
  6. Examples
  7. Linear overview
  8. Notational conventions
  9. Acknowledgments
  10. Background on atomic generation, locally constant objects, and monodromy
  11. Recollections on atomic generation
  12. Sheaves with coefficients & tensor products of presentable ∞-categories
  13. Locally constant objects & monodromy
  14. Exit-path ∞-categories
  15. Stratified ∞-topoi & stratified spaces
  16. ...and 28 more sections

Key Result

Theorem 3.1

Figures (1)

  • Figure 1: A non-conical stratification of $\Sph{1}$ is pictured on the right. On the left is a conical refinement of the right-hand stratification.

Theorems & Definitions (260)

  • Definition 2.1: (cf. arXiv:2108.01924)
  • Theorem 3.1: (stability properties of exodromic stratified spaces; \ref{['thm:stability_properties_of_exodomic_stratified_spaces']})
  • Corollary 4.1: (\ref{['prop:properties_of_locally_conically_refineable_stratified_spaces']})
  • Theorem 4.2: (\ref{['thm:subanalytic_stratified_spaces_are_conically_refineable']})
  • Theorem 4.3: (\ref{['thm:algebraic_stratified_spaces_are_conically_refineable_and_categorically_finite']})
  • Remark 4.4
  • Theorem 4.5: (\ref{['cor:examples_when_ConsP_is_locally_geometric']})
  • Example 5.1
  • Example 5.3: (see \ref{['ex:Favero-Huang_tree_stratification_is_conically_refineable', 'ex:Favero-Huang-Bondal-Ruan_stratification_is_locally_conically_refineable']})
  • Definition 1.1.1
  • ...and 250 more