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Model structure arising from one hereditary complete cotorsion pair on extriangulated categories

Jiangsheng Hu, Dongdong Zhang, Pu Zhang, Panyue Zhou

Abstract

Hovey's correspondence between model structures and cotorsion pairs in the setting of abelian categories, has been generalized by Nakaoka-Palu, using two cotorsion pairs, to the setting of weakly idempotent complete extriangulated categories, and the aim of the paper is to give an analogous correspondence using one (hereditary) cotorsion pair generalizing in this setting work of Beligiannis-Reiten and Cui, Lu and Zhang. Furthermore, we provide methods to construct model structures from silting objects in weakly idempotent complete extriangulated categories and co-$t$-structures on triangulated categories.

Model structure arising from one hereditary complete cotorsion pair on extriangulated categories

Abstract

Hovey's correspondence between model structures and cotorsion pairs in the setting of abelian categories, has been generalized by Nakaoka-Palu, using two cotorsion pairs, to the setting of weakly idempotent complete extriangulated categories, and the aim of the paper is to give an analogous correspondence using one (hereditary) cotorsion pair generalizing in this setting work of Beligiannis-Reiten and Cui, Lu and Zhang. Furthermore, we provide methods to construct model structures from silting objects in weakly idempotent complete extriangulated categories and co--structures on triangulated categories.
Paper Structure (13 sections, 33 theorems, 22 equations)

This paper contains 13 sections, 33 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Extriangulated categories
  4. Cotorsion pairs in extriangulated categories
  5. Model structures
  6. The homotopy category of a model structure
  7. Proof of Theorem \ref{['thmA']}
  8. Model structure induced by a hereditary complete cotorsion pair
  9. Hereditary complete cotorsion pair arising from a model structure
  10. The Beligiannis-Reiten correspondence
  11. Weakly projective model structures
  12. Proof of Theorem \ref{['thm4.6']}
  13. Applications

Key Result

Theorem 1.1

Let $\mathcal{B}$ be a weakly idempotent complete extriangulated category, let $\mathcal{X}$ and $\mathcal{Y}$ be full additive subcategories of $\mathcal{B}$ which are closed under direct summands and isomorphisms, and $\omega=\mathcal{X}\cap \mathcal{Y}$. Then $({\rm CoFib}_\omega, {\rm Fib}_\omeg the class $\mathcal{C}_\omega$ of cofibrant objects is $\mathcal{X}$, the class $\mathcal{F}_\omega

Theorems & Definitions (73)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • ...and 63 more