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Homotopy categories and fibrant model structures

Xue-Song Lu, Pu Zhang

TL;DR

This work identifies the homotopy category of a model structure on a weakly idempotent complete additive category with the additive quotient of cofibrant-fibrant objects by cofibrant-fibrant-trivial objects, formalizing Quillen’s homotopy through localization rather than pushouts/pullbacks. It then provides a precise description of fibrant model structures via trivial cofibrations and via fibrations, and introduces fibrantly weak factorization systems that biject with fibrant model structures. The paper applies these results to recover $\omega$-model structures and $\mathcal{W}$-model structures, clarifying their relations to exact/abelian model structures and giving concrete examples. A dual, cofibrant version is also developed, showing how the homotopy category can be realized as $\mathcal{F}/\mathrm{T}\mathcal{F}$ and establishing a parallel correspondence with cofibrantly weak factorization systems. Overall, the results deepen the understanding of model structures in additive categories beyond exact settings and provide constructive tools for building fibrant (or cofibrant) models from weak factorization data.

Abstract

The homotopy category of a model structure on a weakly idempotent complete additive category is proved to be equivalent to the additive quotient of the category of cofibrant-fibrant objects with respect to the subcategory of cofibrant-fibrant-trivial objects. A model structure on pointed category is fibrant, if every object is a fibrant object. Fibrant model structures is explicitly described by trivial cofibrations, and also by fibrations. Fibrantly weak factorization systems are introduced, fibrant model structures are constructed via fibrantly weak factorization systems, and a one-one correspondence between fibrantly weak factorization systems and fibrant model structures is given. Applications are given to rediscover the $ω$-model structures and the $\mathcal W$-model structures, and their relations with exact model structures are discussed.

Homotopy categories and fibrant model structures

TL;DR

This work identifies the homotopy category of a model structure on a weakly idempotent complete additive category with the additive quotient of cofibrant-fibrant objects by cofibrant-fibrant-trivial objects, formalizing Quillen’s homotopy through localization rather than pushouts/pullbacks. It then provides a precise description of fibrant model structures via trivial cofibrations and via fibrations, and introduces fibrantly weak factorization systems that biject with fibrant model structures. The paper applies these results to recover -model structures and -model structures, clarifying their relations to exact/abelian model structures and giving concrete examples. A dual, cofibrant version is also developed, showing how the homotopy category can be realized as and establishing a parallel correspondence with cofibrantly weak factorization systems. Overall, the results deepen the understanding of model structures in additive categories beyond exact settings and provide constructive tools for building fibrant (or cofibrant) models from weak factorization data.

Abstract

The homotopy category of a model structure on a weakly idempotent complete additive category is proved to be equivalent to the additive quotient of the category of cofibrant-fibrant objects with respect to the subcategory of cofibrant-fibrant-trivial objects. A model structure on pointed category is fibrant, if every object is a fibrant object. Fibrant model structures is explicitly described by trivial cofibrations, and also by fibrations. Fibrantly weak factorization systems are introduced, fibrant model structures are constructed via fibrantly weak factorization systems, and a one-one correspondence between fibrantly weak factorization systems and fibrant model structures is given. Applications are given to rediscover the -model structures and the -model structures, and their relations with exact model structures are discussed.

Paper Structure

This paper contains 26 sections, 30 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. The homotopy category of a model structure
  3. Descriptions of fibrant model structures
  4. Fibrantly weak factorization systems and construction of fibrant model structures
  5. The organization
  6. Preliminaries
  7. Splitting short exact sequences in additive categories
  8. Additive quotient categories
  9. Localization of categories
  10. Model structures
  11. Exact model structures and the Hovey correspondence
  12. Weak factorization systems
  13. The homotopy category
  14. The homotopy categories of model structures
  15. Proof of Theorem \ref{['HoCcapF']}
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $(\operatorname{CoFib}, \operatorname{Fib}, \operatorname{Weq})$ be a model structure on a weakly idempotent complete additive category $\mathcal{A}$. Then the homotopy category ${\rm Ho}(\mathcal{A})$ is equivalent as an additive category to the quotient $(\mathcal{C}\cap \mathcal{F})/(\mathcal

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 35 more