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El. knyga: Abelian Model Category Theory

(Ramapo College of New Jersey)
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Abelian model categories are receiving more attention in contemporary research. Including clear diagrammatic proofs, the book provides an elementary treatment of the foundations of abelian and exact model categories. A self-contained introduction for researchers and graduate students in algebra, topology, representation theory, and category theory.

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.

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