Model categories provide a powerful framework for studying in abstract settings. They come equipped with three classes of morphisms: weak equivalences, fibrations, and cofibrations, which satisfy specific axioms to ensure well-behaved homotopy-theoretic properties.
These categories allow for unified development of homotopy concepts and constructions. They facilitate computations of and enable the study of relationships between different homotopy theories through tools like Quillen adjunctions and equivalences.
Model Categories
Model category axioms
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Define a model category C equipped with three classes of morphisms: weak equivalences, fibrations, and cofibrations
states if f and g are composable morphisms and two of f, g, and gf are weak equivalences, then so is the third
requires the classes of weak equivalences, fibrations, and cofibrations are closed under retracts
specifies cofibrations have the left lifting property with respect to trivial fibrations (fibrations that are also weak equivalences) and fibrations have the right lifting property with respect to trivial cofibrations (cofibrations that are also weak equivalences)
states every morphism f in C can be factored in two ways:
f=pi, where i is a and p is a trivial
f=qj, where j is a trivial cofibration and q is a fibration
Role in homotopy theories
Provide a framework for studying homotopy theories in a general, abstract setting allows development of homotopy-theoretic concepts and constructions in a unified manner
Ensure homotopy-theoretic properties are well-behaved and compatible with the categorical structure through the
Facilitate computations by providing tools for working with homotopy invariants and performing homotopy-theoretic constructions
Lifting properties allow construction of and extensions
Factorization axioms enable computation of and limits (Top, Ch(R))
Examples of model categories
Category of , Top, forms a model category with:
Weak equivalences as weak homotopy equivalences
Fibrations as Serre fibrations
Cofibrations as retracts of
Category of of modules over a ring R, Ch(R), forms a model category with:
Weak equivalences as quasi-isomorphisms (chain maps inducing isomorphisms on )
Fibrations as chain maps that are surjective in positive degrees
Cofibrations as chain maps that are injective with in each degree
Applications to homotopy invariants
Provide a framework for studying homotopy invariants, such as homology and , in a general setting where the axioms ensure these invariants are well-defined and have the expected properties
Allow performing homotopy-theoretic constructions, such as and pullbacks, using the factorization and lifting axioms to construct homotopy colimits and limits
Enable studying the relationship between different homotopy theories through concepts like Quillen adjunctions and Quillen equivalences that compare and relate model categories and transfer homotopy-theoretic information between different settings (Top and Ch(R))