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
6 - The Triangulated Homotopy Category
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 is devoted to studying the triangulated structures that can be attached to an abelian model structure on a weakly idempotent complete exact category. First, one-sided (i.e. left and right) triangulated categories are defined and it is shown that we can attach both a left and right triangulated category to any complete cotorsion pair. It is then shown that the homotopy category of an abelian model structure is always triangulated. In particular, the suspension functor is an autoequivalence of the homotopy category. Natural descriptions of the distinguished triangles are given. Ultimately, it is shown that the canonical functor to the homotopy category may be thought of as the triangulated localization of the exact structure with respect to the class of trivial objects.
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- Abelian Model Category Theory , pp. 117 - 176Publisher: Cambridge University PressPrint publication year: 2025