Recollements and stratification
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-Haine. Combining our theorem with methods from the work of Ayala-Mazel-Gee-Rozenblyum, we then prove a conjecture of Barwick-Glasman-Haine that asserts an equivalence between the $\infty$-category of $P$-stratified $\infty$-topoi and that of toposic locally cocartesian fibrations over $P^{\mathrm{op}}$.
Fields of Study & Subjects
Homotopy and Cohomology in Algebraic Topology
Algebraic structures and combinatorial models
Algebraic Geometry and Number Theory
Publication details
- Journal / source
- Algebraic & Geometric Topology
- Publisher
- Mathematical Sciences Publishers
- Volume
- 26
- Issue
- 4
- Pages
- 1321–1384
- Publication date
- 25 April 2026
- Language
- EN