Table of Contents
Categorical and algebraic preliminaries
Universal constructions
Introduction
The opposite category and duality
Limits and colimits
Abelian categories
Introduction
Additive categories
Abelian categories and exact sequences
Grothendieck categories
Extension functors
Introduction
Projective and injective objects
Projective and injective dimensions
Extension groups via cohomology
Extension groups via Baer description
Applications of Baer extensions: adjunction properties for sphere and disk chain complexes
Torsion functors
Introduction
Monoidal categories
Closed monoidal categories on chain complexes over a ring
Derived functors of -⊗- and Hom(–,–)
Flat chain complexes
Torsion functors and flat dimensions
Interactions between homological algebra and homotopy theory
Model categories
Introduction
Weak factorization systems
Model categories
The homotopy category of a model category
Monoidal model categories
Cotorsion pairs
Introduction
Complete and hereditary cotorsion pairs
Eklof and Trlifaj theorem
Compatible cotorsion pairs
Induced cotorsion pairs of chain complexes
Hovey Correspondence
Introduction
Hovey Correspondence
Abelian factorization systems
Proof of the Hovey Correspondence
Abelian model structures on monoidal categories
Further reading
Classical homological dimensions and abelian model structures on chain complexes
Injective dimensions and model structures
Introduction
n-Injective model structures on chain complexes
Degreewise n-injective model structures on chain complexes
Projective dimensions and model structures
Introduction
Projective dimensions and cotorsion pairs of R-modules
The category Mod(R) of modules over a ringoid
Projective dimensions of modules over ringoids and special precovers
n-projective model structures
Degreewise n-projective model structures
Flat dimensions and model structures
Introduction
The n-flat modules and cotorsion pairs
The n-flat cotorsion pair of chain complexes and model structures
The homotopy category of differential graded model structures
Degreewise n-flat model structures
Further reading
Gorenstein homological dimensions and abelian model structures
Gorenstein-projective and Gorenstein-injective objects
Introduction
Properties of Gorenstein-projective and Gorenstein-injective objects
Gorenstein-projective and Gorenstein-injective dimensions
Gorenstein-injective dimensions and model structures
Introduction
Gorenstein categories
Cotorsion pairs from Gorenstein homological dimensions
Hovey’s model structures on Gorenstein categories
The Gorenstein n-injective model structure
Homotopy categories in Gorenstein homological algebra
Gorenstein-homological dimensions of chain complexes
Further reading
Gorenstein-projective dimensions and model structures
Introduction
G-projective dimensions and cotorsion pairs
Model structures from Gorenstein-projective dimensions
Gorenstein-flat dimensions and model structures
Introduction
Gorenstein-flat modules
Gorenstein-flat dimensions
Gorenstein-flat dimensions of chain complexes
Gorenstein-homological dimensions of graded modules
Further reading
Bibliography
Index
