Introduction to Abelian model structures and Gorenstein homological dimensions / Marco A. Perez.

Includes bibliographical references and index.

I. Categorical and Algebraic Preliminaries:
1. Universal constructions ;
2. Abelian categories ;
3. Extension functors ;
4. Torsion functors --
II.. Interactions Between Homological Algebra and Homotopy Theory:
5. Model categories ;
6. Cotorsion pairs ;
7. Hovey correspondence --
III. Classical Homological Dimensions and Abelian Model Structures on Chain Complexes:
8. Injective dimensions and model structures ;
9. Projective dimensions and model structures ;
10. Flat dimensions and model structures --
IV. Gorenstein Homological Dimensions and Abelian Model Structures:
11. Gorenstein-projective and Gorenstein-injective objects ;
12. Gorenstein-injective dimensions and model structures ;
13. Gorenstein-projective dimensions and model structures ;
14. Gorenstein-flat dimensions and model structures.

This book provides a starting point to study the relationship between homological and homotopical algebra, a very active branch of mathematics. The book shows how to obtain new model structures in homological algebra by constructing a pair of compatible complete cotorsion pairs related to a specific homological dimension and then applying the Hovey Correspondence to generate an abelian model structure.
The first part of the book introduces the definitions and notations of the universal constructions most often used in category theory. The next part presents a proof of the Eklof and Trlifaj theorem in Grothedieck categories and covers M. Hovey’s work that connects the theories of cotorsion pairs and model categories. The final two parts study the relationship between model structures and classical and Gorenstein homological dimensions and explore special types of Grothendieck categories known as Gorenstein categories.
As self-contained as possible, this book presents new results in relative homological algebra and model category theory. The author also re-proves some established results using different arguments or from a pedagogical point of view. In addition, he proves folklore results that are difficult to locate in the literature.