Abelian Model Category Theory

Bridges the gap between traditional methods of homological algebra and Quillen's abstract notion of model category structures.

James Gillespie (Author)

9781009449465, Cambridge University Press

Hardback, published 2 January 2025

438 pages
23.5 x 16 x 3 cm, 0.742 kg

Offering a unique resource for advanced graduate students and researchers, this book treats the fundamentals of Quillen model structures on abelian and exact categories. Building the subject from the ground up using cotorsion pairs, it develops the special properties enjoyed by the homotopy category of such abelian model structures. A central result is that the homotopy category of any abelian model structure is triangulated and characterized by a suitable universal property – it is the triangulated localization with respect to the class of trivial objects. The book also treats derived functors and monoidal model categories from this perspective, showing how to construct tensor triangulated categories from cotorsion pairs. For researchers and graduate students in algebra, topology, representation theory, and category theory, this book offers clear explanations of difficult model category methods that are increasingly being used in contemporary research.

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
A. Hovey's correspondence for general exact categories
B. Right and left homotopy relations
C. Bibliographical notes
References
Index.

Subject Areas: Mathematical foundations [PBC]