Quillen equivalence
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| Quillen equivalence | |
|---|---|
| Name | Quillen equivalence |
| Field | Algebraic topology; Homotopical algebra |
| Introduced by | Daniel Quillen |
| Year | 1967 |
Quillen equivalence is a notion in homotopical algebra that identifies when two model categories present the same homotopy theory. It formalizes equivalence between homotopy categories arising from different model structures and is central to comparisons among constructions in algebraic topology, category theory, and algebraic geometry.
Definition and basic properties
A Quillen equivalence is defined between two model categories equipped with a pair of adjoint functors where the left adjoint is a Quillen left functor and the right adjoint is a Quillen right functor; the unit and counit induce equivalences on the associated homotopy categories. Important contributors to the development of the formalism include Daniel Quillen, André Joyal, Jacob Lurie, Mark Hovey, and Philip S. Hirschhorn. Foundational examples and structural results are discussed in works connected to Quillen model category, model category, simplicial model category, stable model category, and cofibrantly generated model category. Basic properties relate to preservation of weak equivalences, cofibrations, and fibrations, and to derived functors used in proving equivalences; these techniques appear in literature linked to Derived category (triangulated), Brown representability theorem, and Dwyer–Kan simplicial localization.
Examples and important cases
Standard examples include the Quillen equivalences between categories employed by researchers such as G. W. Whitehead and John Milnor, for instance between simplicial sets and topological spaces via the geometric realization and singular complex adjunction appearing in treatments by J. P. May and Serre; comparisons between chain complexes and module spectra in works by Daniel Dugger and Michael Bousfield; and equivalences involving symmetric spectra studied by Stefan Schwede and Mark Hovey. Other pivotal cases arise in the comparison of model structures on categories of operads and algebras treated by Vladimir Hinich, Bertrand Toën, and Gabriele Vezzosi, and in equivalences used in derived algebraic geometry developed by Jacob Lurie, Alexander Grothendieck-inspired programs, and Bertrand Toën. Classical algebraic examples connect differential graded categories (authors like Bernhard Keller), differential graded algebras treated by Henning Krause, and A∞-categories appearing in works connected to Maxim Kontsevich.
Construction and criteria for Quillen equivalences
Constructions often begin with a Quillen adjunction; criteria for upgrading to a Quillen equivalence involve checking that the derived unit and counit are equivalences on cofibrant and fibrant replacements respectively. Techniques draw on localization methods from Dwyer–Kan, small object arguments associated with Jean-Pierre Serre-inspired factorizations, and recognition results by Hovey and Smith-style existence theorems. Practical criteria use homotopy mapping spaces and enriched hom-objects in contexts developed by Charles Rezk, André Joyal, and Jacob Lurie; comparison results also employ tools from the theory of presentable categories advanced by Adámek and Jiri Rosicky and from higher category theory as in work by Emily Riehl.
Homotopy categories and derived functors
A Quillen equivalence induces an equivalence between homotopy categories obtained by inverting weak equivalences; this passage is documented in contexts treated by Verdier and in triangulated settings by Alexandre Grothendieck-influenced authors. Derived functors—left derived and right derived—mediate the passage and are central in expositions by Cartan and Eilenberg as well as modern texts by Hovey and Dugger. The interaction with triangulated and stable homotopy categories features in work by Ivanov, Neeman, and Margolis and has analogues in the study of derived Morita theory developed by Toën and Keller.
Applications in algebraic topology and homotopical algebra
Quillen equivalences enable comparison of homotopical models arising in classical algebraic topology (examples tied to Henri Poincaré, Henri Cartan, Jean Leray traditions), homological algebra, and modern derived geometry. They are employed to transfer structures such as model structures on categories of spectra used by Adams and Boardman, to relate factorization homology frameworks associated with Kevin Costello and Owen Gwilliam, and to compare modular representation theoretic models investigated by Jon F. Carlson. In derived algebraic geometry and motivic homotopy theory, Quillen equivalences underpin comparisons between motivic models considered by Vladimir Voevodsky, Fabien Morel, and Jacob Lurie.
Variants and related notions (e.g., Quillen adjunctions, DK-equivalences)
Related notions include Quillen adjunctions (the starting point), derived equivalences in triangulated settings, Dwyer–Kan equivalences (DK-equivalences) between simplicial categories introduced by W. G. Dwyer and Daniel Kan, and categorical equivalences in higher category theory by André Joyal and Jacob Lurie. Other variants involve Bousfield localizations attributed to A. K. Bousfield, monoidal Quillen equivalences studied by Stefan Schwede and Mark Hovey, and Morita-type equivalences in derived and DG contexts explored by Bernhard Keller and Bertrand Toën.