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
9 - Constructing Complete Cotorsion Pairs
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
The goal of this chapter is to develop a very general method for constructing (functorially) complete cotorsion pairs in exact categories. In essence we develop an algebraic version of Quillen’s small object argument for cotorsion pairs. This naturally leads us to the notion of cofibrantly generated cotorsion pairs and abelian model structures. The approach taken here is inspired by Saorin and Stovicek’s notion of an efficient exact category. We generalize this idea a bit more by considering classes of objects that we say are efficient relative to the exact structure. We show that with mild hypotheses on the exact category, any efficient set (not a proper class) of objects cogenerates a functorially complete cotorsion pair. Along the way we prove results about generators for exact categories, and use them to construct generating monics for cotorsion pairs. We also prove Eklof’s Lemma, and its dual, for exact categories, and give general conditions guaranteeing that the left side of a cotorsion pair is closed under direct limits.
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- Abelian Model Category Theory , pp. 255 - 304Publisher: Cambridge University PressPrint publication year: 2025