Recollements and stratification

Shah, Jay

doi:10.2140/agt.2026.26.1321

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Recollements and stratification

Jay Shah

Algebraic & Geometric Topology 26 (2026) 1321–1384

DOI: 10.2140/agt.2026.26.1321

Abstract

We develop various aspects of the theory of recollements of $\infty$ -categories, including a symmetric monoidal refinement of the theory. Our main result establishes a formula for the gluing functor of a recollement on the right-lax limit of a locally cocartesian fibration determined by a sieve-cosieve decomposition of the base. As an application, we prove a reconstruction theorem for sheaves in an $\infty$ -topos stratified over a finite poset $P$ in the sense of Barwick, Glasman, and Haine. Combining our theorem with methods of Ayala, Mazel-Gee, and Rozenblyum, we then prove a conjecture of Barwick, Glasman, and Haine that asserts an equivalence between the $\infty$ -category of $P$ -stratified $\infty$ -topoi and that of toposic locally cocartesian fibrations over $P^{op}$ .

Keywords

recollements, stratified categories, higher category theory, higher topos theory, reconstruction of sheaves

Mathematical Subject Classification

Primary: 18N60

References

Publication

Received: 17 January 2022

Revised: 19 October 2024

Accepted: 19 March 2025

Published: 25 April 2026

Authors

Jay Shah
Fachbereich Mathematik und Informatik University of Münster Münster Germany

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Volume 26, issue 4 (2026)