Quillen model category
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| Quillen model category | |
|---|---|
| Name | Quillen model category |
| Introduced | 1967 |
| Founder | Daniel Quillen |
| Discipline | Algebraic topology |
Quillen model category
A Quillen model category is an abstract framework introduced to axiomatize homotopy theory within a categorical setting. It provides classes of morphisms — cofibrations, fibrations, and weak equivalences — that interact to permit homotopy-theoretic constructions such as homotopy limits, homotopy colimits, and derived functors. The notion was formalized by Daniel Quillen to unify constructions across contexts including topological spaces, chain complexes, and simplicial sets, and it underlies modern approaches in stable homotopy theory, algebraic K-theory, and motivic homotopy theory.
Definition and basic properties
A model category is a complete and cocomplete category equipped with three distinguished classes of morphisms: cofibrations, fibrations, and weak equivalences, satisfying axioms that guarantee the existence of factorizations and lifting properties. The axioms ensure that any morphism factors functorially as a cofibration followed by a trivial fibration and dually as a trivial cofibration followed by a fibration, which relates to notions arising in Serre spectral sequence contexts and constructions used by Henri Cartan and Samuel Eilenberg. Retracts and two-out-of-three properties mirror structures found in the work of Alexander Grothendieck and are crucial for homotopical manipulations in settings influenced by Jean-Pierre Serre and Samuel Eilenberg.
Examples
Classical examples include the model structure on the category of topological spaces used in Algebraic Topology courses, the Kan model structure on simplicial sets studied by Daniel Kan, and projective and injective model structures on categories of chain complexes over a ring as in work by Jean-Louis Loday and Henning Krause. Further examples appear in categories of dg-algebras, spectra such as those constructed by Mark Hovey and Jeff Smith, and model structures on categories of presheaves used in motivic homotopy theory developed by Fabien Morel and Vladimir Voevodsky.
Constructions and variants
Standard constructions include left and right transferred model structures along adjoint functors, Bousfield localization, and the formation of homotopy limits and colimits. Variants of the notion include combinatorial model categories studied by Jeff Smith and J. Rosický, monoidal model categories used in tensor product contexts such as in work by Schwede-Shipley, and enriched model categories over monoidal model categories as in treatments by Max Kelly and Shulman. Stable model categories, which model stable homotopy categories like those of Boardman and Adams spectral sequence contexts, arise via stabilization procedures in the literature of Paul Goerss and John Jardine.
Homotopy category and derived functors
The homotopy category of a model category is obtained by formally inverting weak equivalences; this construction generalizes classical passage to homotopy classes as in the work of Whitehead and Hurewicz. Derived functors between model categories are obtained by composing functors with cofibrant or fibrant replacement functors, paralleling constructions in homological algebra by Henri Cartan and Samuel Eilenberg. Quillen adjunctions and Quillen equivalences provide criteria for when adjoint functors induce equivalences of homotopy categories, a concept central to comparisons made by Daniel Quillen himself and later developments by Mark Hovey and Philip Hirschhorn.
Cofibrantly generated and combinatorial model categories
Cofibrantly generated model categories are those where sets of generating cofibrations and generating trivial cofibrations permit the small object argument to produce functorial factorizations; this technique traces to work of Michael Barr and Peter Freyd and was systematized in homotopy-theoretic contexts by Frank Adams. Combinatorial model categories combine cofibrant generation with accessibility conditions from accessible category theory developed by Michael Makkai and Jiří Rosický, allowing powerful existence and localization theorems used by Cisinski and Lurie in higher-categorical settings.
Localization and Bousfield localization
Localization of model categories, particularly Bousfield localization introduced by A.K. Bousfield, constructs new model structures in which a chosen class of maps becomes weak equivalences, enabling techniques analogous to localization in algebraic topology and stable homotopy theory. Bousfield localization plays a central role in constructing localized categories such as p-localization of spectra and in the analysis of chromatic homotopy theory by researchers including Douglas Ravenel and Mark Hovey.
Applications and relations to other homotopy theories
Model categories serve as bridges between classical homotopy theory and modern frameworks: they connect to infinity-category approaches of Joyal and Jacob Lurie, to derivator theory of Grothendieck and Maltsiniotis, and to A∞-algebra and E∞-algebra contexts studied by Stasheff and May. Applications span computations in algebraic K-theory influenced by Quillen (K-theory) foundations, structural results in stable homotopy theory used by Adams and Ravenel, and foundations for motivic cohomology as developed by Voevodsky and Morel.