On the triangulated structure of stable monomorphism categories

Jonas Frank; Mathias Schulze

arxiv: 2502.20942 · v2 · submitted 2025-02-28 · 🧮 math.CT · math.RT

On the triangulated structure of stable monomorphism categories

Jonas Frank , Mathias Schulze This is my paper

Pith reviewed 2026-05-23 02:15 UTC · model grok-4.3

classification 🧮 math.CT math.RT

keywords stable monomorphism categoryFrobenius categorytriangulated categorysemiorthogonal decompositionrecollementmutationsuspension functorlinear quiver

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The pith

Symmetry of linear quivers induces polygons of recollements in stable monomorphism categories where a full mutation cycle is a power of an auto-equivalence whose suitable power equals the square of the suspension functor.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that stable monomorphism categories over Frobenius categories with linear quivers possess a triangulated structure shaped by the quiver symmetry. This symmetry produces many semiorthogonal decompositions into smaller categories of the same type. The decompositions close into polygons of recollements. Traversing one full polygon applies a distinguished auto-equivalence, and raising that auto-equivalence to an appropriate power recovers the square of the suspension functor. The same polygons yield infinite chains of adjoint pairs and make the Bondal-Kapranov lifting of representing objects explicit.

Core claim

In the stable monomorphism category of a Frobenius category whose underlying quiver is linear, the symmetry produces semiorthogonal decompositions that organize into polygons of recollements; a complete circuit through any such polygon corresponds to the action of a particular auto-equivalence, and a suitable power of this auto-equivalence coincides with the square of the suspension functor.

What carries the argument

polygons of recollements arising from semiorthogonal decompositions in the stable monomorphism category

Load-bearing premise

The high degree of symmetry of linear quivers is what produces the semiorthogonal decompositions and the closed polygons of recollements.

What would settle it

Explicit computation, for the quiver with two or three vertices, of the composition of mutation functors around one polygon and direct comparison of the result to the square of the suspension functor.

read the original abstract

We investigate the triangulated structure of stable monomorphism categories (filtered chain categories) over a Frobenius category. The high degree of symmetry of linear quivers leads to a plethora of semiorthogonal decompositions into smaller categories of the same type. These form polygons of recollements, in which a full turn of mutations is a power of a particular auto-equivalence of the stable monomorphism category. A certain power of this auto-equivalence is the square of the suspension functor. We describe the infinite chains of adjoint pairs obtained from the polygons. As an application, we explicate the construction of Bondal and Kapranov for lifting representing objects of dualized hom-functors in our setup.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper's main contribution is a description of mutation polygons in stable monomorphism categories over linear quivers whose full cycle yields an auto-equivalence powering to the square of suspension, plus the resulting infinite adjoint chains.

read the letter

The key point is that for these categories the mutations close into polygons of recollements, and one full turn produces an auto-equivalence whose power equals the square of the suspension. The paper also spells out the infinite adjoint chains that come out of the polygons and uses the setup to make the Bondal-Kapranov lifting explicit in this case. That relation between the cycle and suspension is the new structural fact not already in the cited work on stable monomorphism categories. The construction exploits the symmetry of linear quivers to generate many semiorthogonal decompositions that fit together neatly, which is a concrete handle for computation in this narrow setting. The claims line up with standard recollement and mutation methods, and the stress-test finds no circularity or hidden fitting. The obvious limitation is that the whole story depends on the high symmetry of linear quivers, so it stays inside that case and does not claim broader generality. The abstract-only initial review leaves the detailed proofs unchecked, but the internal logic holds without contradiction. This is for people already working on triangulated structures coming from Frobenius categories or quiver representations. A reader who knows the recollement and mutation literature will see the value in the explicit polygons and adjoint chains. It is worth sending to a serious referee because the result is new, the constructions are reproducible in principle, and the scope is clearly stated.

Referee Report

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Summary. The manuscript investigates the triangulated structure of stable monomorphism categories (filtered chain categories) over a Frobenius category. The high degree of symmetry of linear quivers is used to produce a plethora of semiorthogonal decompositions into smaller categories of the same type. These decompositions form polygons of recollements in which a full turn of mutations corresponds to a power of a particular auto-equivalence of the stable monomorphism category; a certain power of this auto-equivalence equals the square of the suspension functor. The paper also describes the infinite chains of adjoint pairs obtained from the polygons and applies the setup to explicate the Bondal-Kapranov construction for lifting representing objects of dualized hom-functors.

Significance. If the constructions and relations hold, the work supplies explicit descriptions of mutation cycles and auto-equivalences in stable categories arising from Frobenius categories with linear quivers, together with concrete infinite adjoint chains and an application to the Bondal-Kapranov lifting procedure. These results would add to the toolkit for analyzing symmetries and recollements in triangulated categories, building on standard techniques while exploiting quiver symmetry to close mutation polygons.

Simulated Author's Rebuttal

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We thank the referee for their summary of the manuscript and for noting its potential contributions to the study of symmetries and recollements in triangulated categories arising from Frobenius categories. The recommendation is listed as uncertain, but the report contains no specific major comments or questions. We therefore have no individual points to address point-by-point at this stage. We remain available to provide clarifications, additional details, or revisions should any concerns be raised.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained categorical construction

full rationale

The paper derives properties of stable monomorphism categories over Frobenius categories using standard recollement and mutation techniques in triangulated categories. The central claims rely on the symmetry of linear quivers to produce semiorthogonal decompositions and polygons of recollements, with mutation cycles relating to the suspension functor via known auto-equivalences. These steps invoke external constructions (e.g., Bondal-Kapranov) without reducing to self-definitions, fitted parameters, or load-bearing self-citations. No equations equate a prediction to its input by construction, and the derivation chain remains independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the existence of a Frobenius structure (standard in the field) and on the symmetry properties of linear quivers; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption The underlying category is Frobenius, hence admits a stable category that is triangulated.
Invoked in the first sentence of the abstract to define the stable monomorphism category.
domain assumption Linear quivers possess sufficient symmetry to generate multiple semiorthogonal decompositions of the same type.
Stated explicitly in the abstract as the source of the polygons of recollements.

pith-pipeline@v0.9.0 · 5636 in / 1409 out tokens · 30432 ms · 2026-05-23T02:15:35.726776+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean; Cost/FunctionalEquation.lean reality_from_one_distinction; washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

polygons of recollements … full turn of mutations is a power of a particular auto-equivalence … Θ^{l+2} ≃ Σ²
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking (D=3) unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

high degree of symmetry of linear quivers … semiorthogonal decompositions … (2l+4)-gons

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.