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arxiv: 2406.18091 · v2 · pith:MS4DC6S3new · submitted 2024-06-26 · 🧮 math.KT · math.CT· math.RT

Weak Waldhausen categories and a localization theorem

Yasuaki Ogawa , Amit Shah This is my paper

Pith reviewed 2026-05-24 00:26 UTC · model grok-4.3

classification 🧮 math.KT math.CTmath.RT
keywords weak Waldhausen categoriesextriangulated categorieslocalization theoremGrothendieck groupalgebraic K-theoryright exact sequencesone-sided localization
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The pith

Weak Waldhausen categories permit one-sided extriangulated localization for right exact Grothendieck group sequences.

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

The authors modify Waldhausen's axioms to create weak Waldhausen categories that align with the structure of extriangulated categories. These categories encompass all extriangulated, exact, and triangulated categories as well as standard Waldhausen categories. The key advantage is support for one-sided localization, which produces right exact sequences of Grothendieck groups unavailable from existing theory. They demonstrate this with new proofs of localization theorems and applications to cluster tilting subcategories and cotorsion pair constructions.

Core claim

A weak Waldhausen category is defined by axioms compatible with any extriangulated category, enabling a one-sided localization theorem that yields right exact sequences of Grothendieck groups.

What carries the argument

The weak Waldhausen category, whose modified axioms allow one-sided extriangulated localization.

If this is right

  • A new proof of the Extriangulated Localization Theorem of Enomoto--Saito at the K0 level.
  • An isomorphism between the K0 of an n-cluster tilting subcategory and the Grothendieck group of an extriangulated substructure.
  • A K0-generalization of Sarazola's localization construction for cotorsion pairs that permits non-Serre localizations.
  • The right exact sequences from the new construction agree with those from the Extriangulated Localization Theorem in common setups.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may allow computation of Grothendieck groups in categories where only one-sided exactness holds.
  • Similar modifications could extend to other K-theory constructions beyond localization.
  • This structure might unify localization results across exact, triangulated, and extriangulated settings.

Load-bearing premise

The modified axioms remain compatible with extriangulated category structure while still supporting the one-sided localization theorem.

What would settle it

A specific extriangulated category equipped with a one-sided localization where the induced map on Grothendieck groups fails to be right exact.

Figures

Figures reproduced from arXiv: 2406.18091 by Amit Shah, Yasuaki Ogawa.

Figure 1
Figure 1. Figure 1: Connections between the key ideas in paper. 1.1. Notation and conventions. Throughout this article we adopt the following conventions. For any category C, we use the following notation. • C(A, B) is the class of morphisms A → B for objects A, B ∈ C . • C → is the class of all morphisms in C . • C →→ is the subclass of C → × C → consisting of all pairs (f, g) of composable morphisms, indicated by diagrams A… view at source ↗
read the original abstract

Waldhausen categories were introduced to extend algebraic $K$-theory beyond Quillen's exact categories. In this article, we modify Waldhausen's axioms so that it matches better with the theory of extriangulated categories, introducing a weak Waldhausen category and defining its Grothendieck group. Examples of weak Waldhausen categories include any extriangulated category, hence any exact or triangulated category, and any Waldhausen category. A key feature of this structure is that it allows for "one-sided" extriangulated localization theory, and thus enables us to extract right exact sequences of Grothendieck groups that we cannot obtain from the theory currently available. To demonstrate the utility of our Weak Waldhausen Localization Theorem, we give three applications. First, we give a new proof of the Extriangulated Localization Theorem proven by Enomoto--Saito, which is a generalization at the level of $K_0$ of Quillen's classical Localization Theorem for exact categories. Second, we give a new proof that the index with respect to an $n$-cluster tilting subcategory $\mathscr{X}$ of a triangulated category $\mathscr{C}$ induces an isomorphism between $K_0^{\mathsf{sp}}(\mathscr{X})$ and the Grothendieck group of an extriangulated substructure of $\mathscr{C}$. Last, we produce a weak Waldhausen $K_0$-generalization of a localization construction due to Sarazola that involves cotorsion pairs but allows for non-Serre localizations. We show that the right exact sequences of Grothendieck groups obtained from our Sarazola construction and the Extriangulated Localization Theorem agree under a common 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.

Referee Report

0 major / 3 minor

Summary. The manuscript defines weak Waldhausen categories by relaxing Waldhausen's axioms to be compatible with extriangulated categories (including exact and triangulated ones). It associates a Grothendieck group to such categories and proves a Weak Waldhausen Localization Theorem yielding right-exact sequences of Grothendieck groups via one-sided localization. Three applications are presented: a new proof of the Enomoto–Saito Extriangulated Localization Theorem, an isomorphism K_0^{sp}(X) ≅ K_0 of an extriangulated substructure induced by an n-cluster tilting subcategory, and a K_0-generalization of Sarazola's cotorsion-pair localization that agrees with the Extriangulated Localization Theorem under a common setup.

Significance. If the central theorem holds, the construction supplies a uniform setting for one-sided extriangulated localization that produces right-exact K_0 sequences unavailable from prior Waldhausen or extriangulated localization theories. The three applications recover known results and extend a cotorsion-pair construction, indicating the framework is broad enough to unify several strands of K-theory localization.

minor comments (3)
  1. [§2] The abstract and introduction state that every extriangulated category is a weak Waldhausen category, but the verification of the relaxed axioms against the extriangulation axioms (e.g., the precise form of the pushout and pullback conditions) should be written out explicitly rather than left as an exercise for the reader.
  2. Notation for the Grothendieck group of a weak Waldhausen category is introduced without a dedicated comparison table to the classical K_0 of exact or Waldhausen categories; adding such a table would clarify the distinction between the new right-exact sequences and previously known exact sequences.
  3. [Application 2] In the cluster-tilting application, the statement that the index map induces an isomorphism should cite the precise definition of the extriangulated substructure on C used to define the target Grothendieck group.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report correctly identifies the main contributions of the paper. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is definition-driven and self-contained

full rationale

The paper defines weak Waldhausen categories by modifying Waldhausen axioms to align with extriangulated categories, then derives the one-sided localization theorem and right-exact K0 sequences directly from those axioms. Examples (extriangulated, exact, triangulated, and standard Waldhausen categories) are shown to satisfy the new axioms by direct verification. The three applications are presented as consequences of the new theorem rather than reductions to prior fitted quantities or self-citations. No load-bearing step equates a claimed result to its own inputs by construction, and external citations (e.g., Enomoto–Saito) are to independent prior work. This is standard axiom-driven mathematics with no detectable circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the newly introduced definition of weak Waldhausen categories and the proof of the localization theorem; these are ad hoc to the paper and not drawn from prior literature.

axioms (1)
  • ad hoc to paper The relaxed axioms for weak Waldhausen categories are compatible with every extriangulated category.
    This compatibility is invoked to ensure the structure applies to exact, triangulated, and extriangulated categories as stated in the abstract.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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    S_•-construction on stable proto-Waldhausen squares categories produces 2-Segal spaces.

Reference graph

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