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Proper morphisms of $\infty$-topoi

Louis Martini, Sebastian Wolf

TL;DR

The paper provides a complete characterization of proper geometric morphisms of ∞-topoi in terms of a relativised compactness relative to a base ∞-topos $\,\mathcal{B}\,$. It develops the theory of $\,\mathcal{B}\,$-categories and internal higher category methods to define compactness via internal filtered colimits and proves that such compact morphisms are precisely the proper ones. The results yield a broad range of applications, including the demonstration that proper and separated maps of topological spaces induce proper morphisms of associated sheaf ∞-topoi, and they extend to coefficients in compactly generated presentable ∞-categories, establishing an $\,\mathcal{E}\,$-linear analogue of proper base change. Collectively, these advances unify base-change properties with internal notions of compactness in a flexible topos-theoretic framework and pave the way for new applications in topology and related areas of geometry.

Abstract

We characterise proper morphisms of $\infty$-topoi in terms of a relativised notion of compactness: we show that a geometric morphism of $\infty$-topoi is proper if and only if it commutes with colimits indexed by filtered internal $\infty$-categories in the target. In particular, our result implies that for any $\infty$-topos, the global sections functor is proper if and only if it preserves filtered colimits. As an application, we show that every proper and separated map of topological spaces gives rise to a proper morphism between the associated sheaf $\infty$-topoi, generalising a result of Lurie. Along the way, we develop some aspects of the theory of localic higher topoi internal to an $\infty$-topos, which might be of independent interest.

Proper morphisms of $\infty$-topoi

TL;DR

The paper provides a complete characterization of proper geometric morphisms of ∞-topoi in terms of a relativised compactness relative to a base ∞-topos . It develops the theory of -categories and internal higher category methods to define compactness via internal filtered colimits and proves that such compact morphisms are precisely the proper ones. The results yield a broad range of applications, including the demonstration that proper and separated maps of topological spaces induce proper morphisms of associated sheaf ∞-topoi, and they extend to coefficients in compactly generated presentable ∞-categories, establishing an -linear analogue of proper base change. Collectively, these advances unify base-change properties with internal notions of compactness in a flexible topos-theoretic framework and pave the way for new applications in topology and related areas of geometry.

Abstract

We characterise proper morphisms of -topoi in terms of a relativised notion of compactness: we show that a geometric morphism of -topoi is proper if and only if it commutes with colimits indexed by filtered internal -categories in the target. In particular, our result implies that for any -topos, the global sections functor is proper if and only if it preserves filtered colimits. As an application, we show that every proper and separated map of topological spaces gives rise to a proper morphism between the associated sheaf -topoi, generalising a result of Lurie. Along the way, we develop some aspects of the theory of localic higher topoi internal to an -topos, which might be of independent interest.
Paper Structure (14 sections, 29 theorems, 17 equations)

This paper contains 14 sections, 29 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Recollection of $\mathcal{B}$-category theory
  3. Main definitions and examples
  4. Left fibrations and Yoneda's lemma
  5. Adjunctions and colimits
  6. Cocompleteness and cocontinuity
  7. Presentable $\mathcal{B}$-categories and $\mathcal{B}$-topoi
  8. Proper geometric morphisms
  9. Compact $\mathcal{B}$-topoi
  10. The toposic cone
  11. Compatibility of pullbacks with localisations
  12. The proof of Theorem \ref{['thm:CompactProperBaseChange']}
  13. Application: proper maps in topology
  14. Proper base change with coefficients

Key Result

Theorem A

Let $\mathcal{X}$ be an $\infty$-topos. Then the global sections functor $\Gamma_{\mathcal{X}}\colon \mathcal{X} \to \mathcal{S}$ is proper if and only if $\mathcal{X}$ is a compact $\infty$-topos.

Theorems & Definitions (89)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 2.1.1: Yoneda
  • Remark 2.1.2
  • Example 2.1.3
  • Definition 2.2.1
  • Proposition 2.2.2: Yoneda
  • Proposition 2.2.3: Yoneda
  • ...and 79 more