Recollements and stratification
Pith reviewed 2026-05-24 12:20 UTC · model grok-4.3
The pith
A formula for the gluing functor in recollements from sieve-cosieve decompositions yields a reconstruction theorem for stratified sheaves and proves an equivalence of P-stratified infinity-topoi with toposic locally cocartesian fibrations.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The 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, this yields a reconstruction theorem for sheaves in an infinity-topos stratified over a finite poset P. Combining the theorem with other methods then proves that the infinity-category of P-stratified infinity-topoi is equivalent to the infinity-category of toposic locally cocartesian fibrations over P^op.
What carries the argument
The 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.
If this is right
- Sheaves on a P-stratified infinity-topos are recoverable from their restrictions to the strata together with the gluing data supplied by the formula.
- The infinity-category of P-stratified infinity-topoi is equivalent to the infinity-category of toposic locally cocartesian fibrations over P^op.
- The theory of recollements admits a symmetric monoidal refinement.
Where Pith is reading between the lines
- The equivalence supplies two interchangeable models for working with stratified infinity-topoi, one based on stratification data and one based on fibrations.
- The gluing formula may support explicit calculations of limits or colimits inside these categories once the models are identified.
Load-bearing premise
The gluing formula combines with techniques from related work to establish both the reconstruction theorem and the full equivalence for finite posets.
What would settle it
A direct computation for a two-point poset showing that the gluing functor fails to match the map predicted by the formula would falsify the central result.
read the original 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 $P^{\mathrm{op}}$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper develops the theory of recollements of ∞-categories, including a symmetric monoidal refinement. Its main result gives an explicit formula for the gluing functor of a recollement on the right-lax limit of a locally cocartesian fibration arising from a sieve-cosieve decomposition of the base. As an application it proves a reconstruction theorem for sheaves in an ∞-topos stratified over a finite poset P (in the sense of Barwick-Glasman-Haine). Combining this with techniques from Ayala-Mazel-Gee-Rozenblyum, the paper establishes an equivalence between the ∞-category of P-stratified ∞-topoi and the ∞-category of toposic locally cocartesian fibrations over P^op, thereby proving a conjecture of Barwick-Glasman-Haine.
Significance. If the central claims hold, the work supplies a concrete computational tool (the gluing-functor formula) and a reconstruction theorem that directly address questions in stratified higher topos theory. The resolution of the Barwick-Glasman-Haine conjecture via an explicit equivalence is a substantive advance; the combination of the new formula with existing methods from Ayala-Mazel-Gee-Rozenblyum yields a falsifiable categorical equivalence that can be checked in low-dimensional cases.
minor comments (3)
- §1.3, Definition 1.3.4: the notation for the right-lax limit is introduced without an explicit comparison to the lax limit used in Ayala-Mazel-Gee-Rozenblyum; a short remark clarifying the variance would aid readers.
- Theorem 4.2.1: the statement of the reconstruction theorem assumes P is finite, but the surrounding discussion does not indicate whether the finiteness hypothesis is essential or merely convenient for the sieve-cosieve argument; a one-sentence remark would clarify the scope.
- Notation section: several diagrams are labeled with the same letter (e.g., 'F') in different sections; consistent subscripting or a global notation table would reduce ambiguity.
Simulated Author's Rebuttal
We thank the referee for the detailed and positive assessment of our manuscript. The recommendation for minor revision is noted, and we will prepare a revised version accordingly. No major comments were raised in the report.
Circularity Check
No significant circularity detected
full rationale
The paper develops recollement theory and states a main result giving an explicit formula for a gluing functor on right-lax limits of locally cocartesian fibrations, followed by a reconstruction theorem for sheaves on finite-poset-stratified ∞-topoi and a proof of the Barwick-Glasman-Haine conjecture obtained by combining the new theorem with external methods from Ayala-Mazel-Gee-Rozenblyum. No load-bearing step reduces by definition, by fitted-parameter renaming, or by self-citation chain to the paper's own inputs; all cited prior work is by distinct authors and is presented as independent input. The derivation chain is therefore self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Standard axioms and properties of infinity-categories, recollements, and locally cocartesian fibrations
- domain assumption Stratification of an infinity-topos over a finite poset P in the sense of Barwick-Glasman-Haine
Forward citations
Cited by 1 Pith paper
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The Galois theory of $G$-spectra and the Burnside ring
The Galois groupoid of G-spectra is equivalent to the étale fundamental groupoid of the Burnside ring of G.
Reference graph
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discussion (0)
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