Abelian Model Category Theory

James Gillespie (Ramapo College of New Jersey)

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English

Cambridge University Press
02 January 2025

Mathematical logic; Set theory; Geometry; Topology

Series: Cambridge Studies in Advanced Mathematics

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.

By: James Gillespie (Ramapo College of New Jersey)
Imprint: Cambridge University Press
Country of Publication: United Kingdom
ISBN: 9781009449465
ISBN 10: 100944946X
Series: Cambridge Studies in Advanced Mathematics
Pages: 436
Publication Date: 02 January 2025
Audience: General/trade , ELT Advanced
Format: Hardback
Publisher's Status: Active

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.

James Gillespie is Professor of Mathematics at Ramapo College of New Jersey. His research interests are homological algebra and abstract homotopy theory and he is the author of thirty-five well-cited articles in the area, particularly on topics such as rings and modules, chain complexes, and sheaves.