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Localizing invariants of constructible sheaves

Qingyuan Bai, Peter J. Haine

Abstract

Given an open-closed decomposition of the stratifying poset, we construct a new semi-orthogonal decomposition of the $\infty$-category of constructible sheaves on a stratified space admitting an exit-path $\infty$-category. From this we obtain a direct sum decomposition of the localizing invariants of the $\infty$-category of constructible sheaves. Since the $\ast$-pullback to the open stratum in the usual (recollement) semi-orthogonal decomposition is not strongly left adjoint, this splitting does not follow from pure sheaf theory considerations. Instead, the splitting crucially relies on the exodromy equivalence: it implies that on the level of constructible sheaves, the $\ast$-pullback to a closed stratum and the $!$-pushforward from an open stratum admit left adjoints. These new functors provide an additional semi-orthogonal decomposition (with the roles of open and closed reversed) in which the relevant functors are strongly left adjoint.

Localizing invariants of constructible sheaves

Abstract

Given an open-closed decomposition of the stratifying poset, we construct a new semi-orthogonal decomposition of the -category of constructible sheaves on a stratified space admitting an exit-path -category. From this we obtain a direct sum decomposition of the localizing invariants of the -category of constructible sheaves. Since the -pullback to the open stratum in the usual (recollement) semi-orthogonal decomposition is not strongly left adjoint, this splitting does not follow from pure sheaf theory considerations. Instead, the splitting crucially relies on the exodromy equivalence: it implies that on the level of constructible sheaves, the -pullback to a closed stratum and the -pushforward from an open stratum admit left adjoints. These new functors provide an additional semi-orthogonal decomposition (with the roles of open and closed reversed) in which the relevant functors are strongly left adjoint.

Paper Structure

This paper contains 9 sections, 19 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Dualizable ∞-categories and localizing invariants
  4. Exodromy
  5. Recollements
  6. Semi-orthogonal decompositions
  7. Splitting results
  8. Large ∞-categories
  9. Small ∞-categories

Key Result

Theorem 2

Let $s \colon \fromto{\Ccal}{P}$ be a functor from a small to a poset. For each $p \in P$, write $\Ccal_p \colonequals s^{-1}(p)$. Then for any dualizable presentable stable $\Ecal$, the functors given by left Kan extension along the inclusions $\Ccal_p \inclusion \Ccal$ induce a natural equivalenc

Theorems & Definitions (46)

  • Theorem 2: (\ref{['cor:splitting_for_functors']})
  • Theorem 3: (\ref{['cor:splitting_for_exodromic_stratified_topoi']})
  • Remark 4: (why exodromy is needed)
  • Corollary 5: (\ref{['rem:deducing_lattice_conjecture_from_strata']})
  • Theorem 6: (\ref{['cor:splitting_constructible_small_category_finite_poset']})
  • Remark 8
  • Example 9
  • Remark 10
  • Remark 11
  • Example 1.6
  • ...and 36 more