Recollements and stratification

TL;DR

This paper advances the theory of recollements in $$-categories, providing a formula for the gluing functor and a reconstruction theorem for stratified sheaves, confirming a conjecture on stratified $$-topoi.

Contribution

It introduces a symmetric monoidal refinement, a formula for the gluing functor, and proves a conjecture relating stratified $$-topoi to locally cocartesian fibrations.

Findings

Established a formula for the gluing functor in recollements.

Proved a reconstruction theorem for sheaves in stratified $$-topoi.

Confirmed a conjecture linking stratified $$-topoi and toposic fibrations.

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 $P^{\mathrm{op}}$.