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Abelian categories from triangulated categories via Nakaoka-Palu's localization

Yasuaki Ogawa

TL;DR

The paper develops a generalized Hovey twin cotorsion pair (gHTCP) localization in extriangulated categories, unifying and extending Abe–Nakaoka’s heart construction and Buan–Marsh/Iyama–Yoshino realizations. It shows that the heart of a cotorsion pair can be realized as a Gabriel-Zisman localization associated to a gHTCP, and that this framework recovers known localizations (including HTCP) and encompasses stable/quotient constructions and recollements of triangulated and abelian categories. A key result is that, under functorial finiteness, the additive quotient by $ olinebreak* olinebreak$ becomes preabelian and the regular morphisms admit a left/right calculus, with the localization factoring through a preabelian stage to yield the abelian heart. The approach provides a unified, universal perspective on constructing abelian categories from triangulated data, clarifying the relationships among subfactors, recollements, and heart constructions, and highlighting the role of stable and localized structures in representation theory. Overall, the work broadens the toolkit for deriving module- and abelian-like categories from triangulated contexts and clarifies their universal properties via gHTCP localization.

Abstract

The aim of this paper is to provide an expansion to Abe-Nakaoka's heart construction of the following two different realizations of the module category over the endomorphism ring of a rigid object in a triangulated category: Buan-Marsh's localization and Iyama-Yoshino's subfactor. Our method depends on a modification of Nakaoka-Palu's HTCP localization, a Gabriel-Zisman localization of extriangulated categories which is also realized as a subfactor of the original ones. Besides of the heart construction, our generalized HTCP localization involves the following phenomena: (1) stable category with respect to a class of objects; (2) recollement of triangulated categories; (3) recollement of abelian categories under a mild assumption.

Abelian categories from triangulated categories via Nakaoka-Palu's localization

TL;DR

The paper develops a generalized Hovey twin cotorsion pair (gHTCP) localization in extriangulated categories, unifying and extending Abe–Nakaoka’s heart construction and Buan–Marsh/Iyama–Yoshino realizations. It shows that the heart of a cotorsion pair can be realized as a Gabriel-Zisman localization associated to a gHTCP, and that this framework recovers known localizations (including HTCP) and encompasses stable/quotient constructions and recollements of triangulated and abelian categories. A key result is that, under functorial finiteness, the additive quotient by becomes preabelian and the regular morphisms admit a left/right calculus, with the localization factoring through a preabelian stage to yield the abelian heart. The approach provides a unified, universal perspective on constructing abelian categories from triangulated data, clarifying the relationships among subfactors, recollements, and heart constructions, and highlighting the role of stable and localized structures in representation theory. Overall, the work broadens the toolkit for deriving module- and abelian-like categories from triangulated contexts and clarifies their universal properties via gHTCP localization.

Abstract

The aim of this paper is to provide an expansion to Abe-Nakaoka's heart construction of the following two different realizations of the module category over the endomorphism ring of a rigid object in a triangulated category: Buan-Marsh's localization and Iyama-Yoshino's subfactor. Our method depends on a modification of Nakaoka-Palu's HTCP localization, a Gabriel-Zisman localization of extriangulated categories which is also realized as a subfactor of the original ones. Besides of the heart construction, our generalized HTCP localization involves the following phenomena: (1) stable category with respect to a class of objects; (2) recollement of triangulated categories; (3) recollement of abelian categories under a mild assumption.

Paper Structure

This paper contains 16 sections, 40 theorems, 13 equations.

Table of Contents

  1. Preliminary
  2. Extriangulated category
  3. Gabriel-Zisman localization
  4. Preabelian category
  5. Localization via generalized Hovey twin cotorsion pair (gHTCP)
  6. Definition of gHTCP
  7. Comparison between gHTCP and HTCP
  8. Examples: recollement of abelian/triangulated categories
  9. Additive quotient
  10. Recollement of triangulated categories
  11. Recollement of abelian categories
  12. Aspects of the heart construction
  13. Basic properties of the heart
  14. From cotorsion pair to gHTCP
  15. A factorization through a preabelian category
  16. ...and 1 more sections

Key Result

Theorem A

Let $((\mathcal{S},\mathcal{T}),(\mathcal{U},\mathcal{V}))$ be a generalized HTCP in an extriangulated category $\mathcal{C}$. Then, there exists a class $\mathbb{V}$ of morphisms in $\mathcal{C}$ such that the associated Gabriel-Zisman localization $\mathsf{L}_\mathbb{V}:\mathcal{C}\to\mathcal{C}[\

Theorems & Definitions (80)

  • Theorem A: Theorem \ref{['thm:RL=LR_implies_universality']}
  • Theorem B: Corollary \ref{['cor:heart_is_gHTCP']}
  • Theorem C: Theorem \ref{['thm:generalized_BM2']}
  • Definition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Definition 1.4
  • Definition 1.5
  • Example 1.6
  • proof
  • ...and 70 more