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3 results

  • 4 - Hovey Triples and Abelian Model Structures

  • James Gillespie, Ramapo College of New Jersey
  • Book:
    Abelian Model Category Theory
    Published online:
    19 December 2024
    Print publication:
    02 January 2025, pp 71-92

  • Summary

    This chapter develops the fundamentals of abelian model structures from the perspective of cotorsion pairs in exact categories. The key notion is that of a Hovey triple. This is a triple of classes of objects which are intertwined to form two complete cotorsion pairs. From a given Hovey triple we define (co)fibrations and weak equivalences as well as the (very good) left and right homotopy relations and their stable categories. The notion of a trivial morphism is introduced and it is shown that the 2 out of 3 property for weak equivalences is equivalent to the statement that each trivial morphism is a weak equivalence. This condition is automatic when the underlying additive category is weakly idempotent complete. At the end of the chapter, Hovey’s correspondence between cotorsion pairs (i.e. Hovey triples) and abelian model structures is proved.

Abelian Model Category Theory
  • James Gillespie
  • Published online:
    19 December 2024
    Print publication:
    02 January 2025
  • 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.