15.3 Homotopy theory and model categories

Model categories provide a powerful framework for studying homotopy theories 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 homotopy invariants 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 $\mathcal{C}$ equipped with three classes of morphisms: weak equivalences, fibrations, and cofibrations
• Two-out-of-three axiom states if $f$ and $g$ are composable morphisms and two of $f$, $g$, and $gf$ are weak equivalences, then so is the third
• Retract axiom requires the classes of weak equivalences, fibrations, and cofibrations are closed under retracts
• Lifting properties axiom 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)
• Factorization axiom states every morphism $f$ in $\mathcal{C}$ can be factored in two ways:
  1. $f = pi$, where $i$ is a cofibration and $p$ is a trivial fibration
  2. $f = qj$, 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 model category axioms
• Facilitate computations by providing tools for working with homotopy invariants and performing homotopy-theoretic constructions
  • Lifting properties allow construction of homotopy lifts and extensions
  • Factorization axioms enable computation of homotopy colimits and limits ($\mathbf{Top}$, $\mathbf{Ch}(R)$)

Examples of model categories

• Category of topological spaces, $\mathbf{Top}$, forms a model category with:
  • Weak equivalences as weak homotopy equivalences
  • Fibrations as Serre fibrations
  • Cofibrations as retracts of relative cell complexes
• Category of chain complexes of modules over a ring $R$, $\mathbf{Ch}(R)$, forms a model category with:
  • Weak equivalences as quasi-isomorphisms (chain maps inducing isomorphisms on homology)
  • Fibrations as chain maps that are surjective in positive degrees
  • Cofibrations as chain maps that are injective with projective cokernel in each degree

Applications to homotopy invariants

• Provide a framework for studying homotopy invariants, such as homology and cohomology, 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 homotopy pushouts 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 ($\mathbf{Top}$ and $\mathbf{Ch}(R)$)