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Categories of Constructible Sheaves

Valery Lunts, Olaf Schnuerer

TL;DR

The paper analyzes when the realization functor from derived categories of constructible/locally constant sheaves on stratified spaces is an equivalence with the corresponding derived category of sheaves having constructible cohomology. It develops a comprehensive framework using Loc, Cons, and stratified categories, and proves a one-stratum derived equivalence under local acyclicity, then extends this to multi-stratum spaces via gluing with 2-morphisms that compare real and pushforward functors. Key contributions include explicit fiber-functor descriptions for Loc, equivariant extensions, and a detailed set of equivalent conditions (including sigma and tau) guaranteeing the global equivalence, together with stratified-normal-structure criteria and concrete instances such as toric varieties. The results yield positive answers for finite-type constructs in several important cases and provide a roadmap for when a full derived-equivalence holds, with explicit counterexamples illustrating limitations. The work advances understanding of constructible and locally constant sheaves in topological and algebraic settings, offering tools for SEO-sensitive summaries and further algorithmic exploration of derived categories in topology and algebraic geometry.

Abstract

Given a stratified topological space, we answer the question whether the functor from the derived category of constructible sheaves to the derived category of sheaves with constructible cohomology is an equivalence. We also establish basic facts on the category of locally constant sheaves and on the category of constructible sheaves.

Categories of Constructible Sheaves

TL;DR

The paper analyzes when the realization functor from derived categories of constructible/locally constant sheaves on stratified spaces is an equivalence with the corresponding derived category of sheaves having constructible cohomology. It develops a comprehensive framework using Loc, Cons, and stratified categories, and proves a one-stratum derived equivalence under local acyclicity, then extends this to multi-stratum spaces via gluing with 2-morphisms that compare real and pushforward functors. Key contributions include explicit fiber-functor descriptions for Loc, equivariant extensions, and a detailed set of equivalent conditions (including sigma and tau) guaranteeing the global equivalence, together with stratified-normal-structure criteria and concrete instances such as toric varieties. The results yield positive answers for finite-type constructs in several important cases and provide a roadmap for when a full derived-equivalence holds, with explicit counterexamples illustrating limitations. The work advances understanding of constructible and locally constant sheaves in topological and algebraic settings, offering tools for SEO-sensitive summaries and further algorithmic exploration of derived categories in topology and algebraic geometry.

Abstract

Given a stratified topological space, we answer the question whether the functor from the derived category of constructible sheaves to the derived category of sheaves with constructible cohomology is an equivalence. We also establish basic facts on the category of locally constant sheaves and on the category of constructible sheaves.
Paper Structure (18 sections, 36 theorems, 120 equations)

This paper contains 18 sections, 36 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Conventions
  3. Formulation of main results
  4. Locally constant sheaves
  5. Constructible sheaves
  6. Realization functor
  7. Local systems of finite rank
  8. Locally constant sheaves: Proof of Proposition \ref{['p:fiber-functor-1']}
  9. $G$-spaces and equivariant sheaves
  10. Proof of Proposition \ref{['p:fiber-functor-1']}
  11. Derived category ${\operatorname{D}}({\operatorname{Loc}}(X))$: proof of Theorem \ref{['t:main-one-stratum-1']}
  12. Constructible sheaves: Proof of Proposition \ref{['p:loc-sc-grothendieck-1']} and Theorem \ref{['t:equivalent-conditions-for-equivalence-1']}
  13. Strategy of the proof of Theorem \ref{['t:equivalent-conditions-for-equivalence-1']}
  14. Completion of the proof of Theorem \ref{['t:equivalent-conditions-for-equivalence-1']}
  15. Stratified spaces with normal structure
  16. ...and 3 more sections

Key Result

Proposition 2.5

Let $X$ be non-empty, path connected, locally simply connected topological space. Let $x \in X$. Then the following holds. (1) There is a natural equivalence of categories Therefore the category ${\operatorname{Loc}} (X)$ is a (complete and co-complete) Grothendieck abelian category. It has enough projectives and enough injectives. (2) The inclusion functor ${\operatorname{Loc}}(X)\hookrightarrow

Theorems & Definitions (86)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.5
  • Theorem 2.7
  • Definition 2.8
  • Definition 2.9
  • Proposition 2.10
  • Definition 2.11
  • Lemma 2.12
  • ...and 76 more