Model categories
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| Model categories | |
|---|---|
| Name | Model categories |
| Discipline | Algebraic topology; Category theory; Homotopy theory |
| Introduced | Quillen (1967) |
| Key concepts | Weak equivalences; Fibrations; Cofibrations; Homotopy category |
| Notable people | Daniel Quillen; J. Peter May; Gideon Maschke; Mark Hovey; Philip S. Hirschhorn |
Model categories
Model categories are axiomatic frameworks introduced to organize homotopy-theoretic ideas across diverse contexts. They formalize how one can invert a class of morphisms up to homotopy and control constructions such as homotopy limits and colimits. Originating in work by Daniel Quillen, the theory has influenced research at the intersection of Algebraic topology, Category theory, Homological algebra, and modern fields like Algebraic geometry and Higher category theory.
Introduction
A model category equips a category with three distinguished classes of morphisms: weak equivalences, fibrations, and cofibrations, satisfying axioms that permit homotopical manipulation. This structure abstracts classical settings such as the category of topological spaces used by Henri Poincaré and L. E. J. Brouwer and chain complexes studied by Jean-Pierre Serre. Model categories enable comparisons between settings via Quillen adjunctions and Quillen equivalences, central in work by Daniel Quillen, Quillen's students, and later expounded by J. Peter May and Mark Hovey.
Definitions and basic concepts
A model category C is a complete and cocomplete category endowed with three classes of maps: weak equivalences W, fibrations Fib, and cofibrations Cof, satisfying axioms (M1)–(M5) as in Quillen's formulation. The 2-out-of-3 property for W, lifting properties characterizing trivial cofibrations and trivial fibrations, and functorial factorization into cofibration followed by trivial fibration or trivial cofibration followed by fibration are required. Fundamental notions include cofibrant and fibrant objects, cylinder and path objects, and homotopy relations defined via cylinder or path data; these ideas connect to constructions by Henri Cartan and techniques used by Samuel Eilenberg and Saunders Mac Lane.
Examples and important model categories
Classical examples include the Serre model structure on the category of compactly generated Hausdorff Topological spaces (as developed by practitioners following Jean-Pierre Serre and Edwin H. Spanier), the Kan–Quillen model structure on simplicial sets related to work by Daniel Kan and Daniel Quillen, and the projective and injective model structures on chain complexes over a ring studied by Jean-Louis Loday and Joseph J. Rotman. Other pivotal instances are model structures on differential graded algebras influenced by Max Karoubi and Henri Cartan, symmetric spectra and S-modules advanced by Mark Hovey, Stefan Schwede, and Shipley, and model structures on operads linked to J. Peter May and Michael Boardman. Model categories also appear in contexts like equivariant homotopy theory (work of Peter May and G. Segal), motivic homotopy theory initiated by Fabien Morel and Vladimir Voevodsky, and derived algebraic geometry developed by Jacob Lurie and Bertrand Toën.
Constructions and methods
Standard constructions include transferred model structures via adjoint functors, left and right Bousfield localizations named after Armand Borel and collaborators in spirit, and cellularization or stabilization procedures. Small object argument, introduced by Daniel Quillen and formalized in later expositions by Philip S. Hirschhorn, gives functorial factorizations using sets of generating cofibrations. Homotopy (co)limits are built using Reedy model structures (as in work by Chris Reedy), bar and cobar constructions related to Eilenberg–Mac Lane techniques, and model categorical enrichments over monoidal model categories studied by Max Kelly and J. Peter May. Techniques for producing model structures on diagram categories and categories of algebras over monads draw on contributions from G. M. Kelly and Johnstone.
Homotopy category and derived functors
Passing to the homotopy category Ho(C) is achieved by formally inverting weak equivalences or by restricting to bifibrant objects and identifying homotopy classes of maps; this reflects earlier localization ideas present in Jean-Pierre Serre's work. Derived functors arise from Quillen adjunctions: a left Quillen functor has a total left derived functor between homotopy categories, and right Quillen functors produce total right derived functors. Quillen equivalences induce equivalences of homotopy categories, a principle used extensively in comparisons such as between simplicial sets and topological spaces (classical Quillen equivalence) or between differential graded categories and spectral categories in modern treatments by Jacob Lurie and Bertrand Toën.
Variations and generalizations
Variations include combinatorial model categories (locally presentable with cofibrantly generated structures) studied by Peter Johnstone and Jeff Smith, cellular model categories with sets of generating cells, and model structures enriched over monoidal model categories such as symmetric spectra explored by Mark Hovey and Stefan Schwede. Higher-categorical analogues, like model structures for (infinity,1)-categories and the Joyal model structure on simplicial sets developed by André Joyal and refined in Jacob Lurie's work, extend the framework. Other generalizations encompass relative model categories, monoidal model categories, and semi-model categories employed in settings with restricted lifting.
Applications and impact on mathematics
Model categories unify and extend homotopical methods across Algebraic topology, Homological algebra, Algebraic geometry, and Mathematical physics. They underpin modern derived and homotopical algebraic geometry as in work by Bertrand Toën, Gabriele Vezzosi, and Jacob Lurie, enable computations in stable homotopy theory via spectra by Mark Hovey and Stefan Schwede, and facilitate operadic and higher-structure analyses guided by J. Peter May and Michael Boardman. Model-categorical tools inform fields as diverse as deformation theory (influenced by Maxim Kontsevich), motivic homotopy (by Vladimir Voevodsky), and the study of ring spectra (by Elmendorf and Mandell), making model categories a central organizing language in contemporary mathematical research.