Introduction to Abelian Model Structures and Gorenstein Homological Dimensions
Chapman & Hall/CRC Monographs and Research Notes in Mathematics Series

Introduction to Abelian Model Structures and Gorenstein Homological Dimensions 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.

Categorical and algebraic preliminaries. Interactions between homological algebra and homotopy theory. Classical homological dimensions and abelian model structures on chain complexes. Gorenstein homological dimensions and abelian model structures. Bibliography. Index.

Dr. Marco A. Pérez is a postdoctoral fellow at the Mathematics Institute of the Universidad Nacional Autónoma de México, where he works on Auslander–Buchweitz approximation theory and cotorsion pairs. He was previously a postdoctoral associate at the Massachusetts Institute of Technology, working on category theory applied to communications and linguistics. Dr. Pérez’s research interests cover topics in both category theory and homological algebra, such as model category theory, ontologies, homological dimensions, Gorenstein homological algebra, finitely presented modules, modules over rings with many objects, and cotorsion theories. He received his PhD in mathematics from the Université du Québec à Montréal in the spring of 2014.