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Model structure from one hereditary complete cortorsion pair

Jian Cui, Xue-Song Lu, Pu Zhang

Abstract

In contrast with the Hovey correspondence of abelian model structures from two complete cotorsion pairs, Beligiannis and Reiten give a construction of model structures on abelian categories from only one complete cotorsion pair. The aim of this paper is to extend this result to weakly idempotent complete exact categories, by adding the condition of heredity of the complete cotorsion pair. In fact, even for abelian categories, this condition of heredity should be added. This construction really gives model structures which are not necessarily exact in the sense of Gillespie. The correspondence of Beligiannis and Reiten of weakly projective model structures also holds for weakly idempotent complete exact categories.

Model structure from one hereditary complete cortorsion pair

Abstract

In contrast with the Hovey correspondence of abelian model structures from two complete cotorsion pairs, Beligiannis and Reiten give a construction of model structures on abelian categories from only one complete cotorsion pair. The aim of this paper is to extend this result to weakly idempotent complete exact categories, by adding the condition of heredity of the complete cotorsion pair. In fact, even for abelian categories, this condition of heredity should be added. This construction really gives model structures which are not necessarily exact in the sense of Gillespie. The correspondence of Beligiannis and Reiten of weakly projective model structures also holds for weakly idempotent complete exact categories.
Paper Structure (32 sections, 34 theorems, 23 equations)

This paper contains 32 sections, 34 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. The $\omega$-model structures
  3. The heredity
  4. The correspondence of Beligiannis and Reiten
  5. The organization
  6. Preliminaries
  7. Exact categories
  8. Extension-Lifting Lemma
  9. Weakly idempotent complete exact categories
  10. Cotorsion pairs in exact categories
  11. Model structures
  12. Quillen's homotopy category
  13. The Hovey correspondence
  14. Model structure induced by a hereditary complete cotorsion pair
  15. Descriptions of ${\rm CoFib}_{\omega} \cap \operatorname{Weq}_{\omega}$ and ${\rm Fib}_{\omega}\cap \operatorname{Weq}_{\omega}$
  16. ...and 17 more sections

Key Result

Theorem 1.1

(See Theorems ifpart and onlyif) Let $\mathcal{A}$ be a weakly idempotent complete exact category, $\mathcal{X}$ and $\mathcal{Y}$ full additive subcategories of $\mathcal{A}$ which are closed under isomorphisms and direct summands, and $\omega =\mathcal{X}\cap \mathcal{Y}$. Then $({\rm CoFib}_{\om

Theorems & Definitions (58)

  • Theorem 1.1
  • Proposition 1.2
  • proof
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • ...and 48 more