Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Introduction and Main Examples
- 1 Additive and Exact Categories
- 2 Cotorsion Pairs
- 3 Stable Categories from Cotorsion Pairs
- 4 Hovey Triples and Abelian Model Structures
- 5 The Homotopy Category of an Abelian Model Structure
- 6 The Triangulated Homotopy Category
- 7 Derived Functors and Abelian Monoidal Model Structures
- 8 Hereditary Model Structures
- 9 Constructing Complete Cotorsion Pairs
- 10 Abelian Model Structures on Chain Complexes
- 11 Mixed Model Structures and Examples
- 12 Cofibrant Generation and Well-Generated Homotopy Categories
- Appendix A Hovey’s Correspondence for General Exact Categories
- Appendix B Right and Left Homotopy Relations
- Appendix C Bibliographical Notes
- References
- Index
4 - Hovey Triples and Abelian Model Structures
Published online by Cambridge University Press: 19 December 2024
- Frontmatter
- Dedication
- Contents
- Preface
- Introduction and Main Examples
- 1 Additive and Exact Categories
- 2 Cotorsion Pairs
- 3 Stable Categories from Cotorsion Pairs
- 4 Hovey Triples and Abelian Model Structures
- 5 The Homotopy Category of an Abelian Model Structure
- 6 The Triangulated Homotopy Category
- 7 Derived Functors and Abelian Monoidal Model Structures
- 8 Hereditary Model Structures
- 9 Constructing Complete Cotorsion Pairs
- 10 Abelian Model Structures on Chain Complexes
- 11 Mixed Model Structures and Examples
- 12 Cofibrant Generation and Well-Generated Homotopy Categories
- Appendix A Hovey’s Correspondence for General Exact Categories
- Appendix B Right and Left Homotopy Relations
- Appendix C Bibliographical Notes
- References
- Index
Summary
This chapter develops the fundamentals of abelian model structures from the perspective of cotorsion pairs in exact categories. The key notion is that of a Hovey triple. This is a triple of classes of objects which are intertwined to form two complete cotorsion pairs. From a given Hovey triple we define (co)fibrations and weak equivalences as well as the (very good) left and right homotopy relations and their stable categories. The notion of a trivial morphism is introduced and it is shown that the 2 out of 3 property for weak equivalences is equivalent to the statement that each trivial morphism is a weak equivalence. This condition is automatic when the underlying additive category is weakly idempotent complete. At the end of the chapter, Hovey’s correspondence between cotorsion pairs (i.e. Hovey triples) and abelian model structures is proved.
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- Abelian Model Category Theory , pp. 71 - 92Publisher: Cambridge University PressPrint publication year: 2025