2020/09/10 by Jaqueline Girabel, Girabel, Jaqueline
Mathematics · #18N10 #18N40 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2009.05390
openalex publication_date 2020/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In [Homotopical Algebra, Springer LNM 43] Quillen introduces the notion of a model category: a category C provided with three distinguished classes of maps \W, F, coF\ (weak equivalences, fibrations, cofibrations), and gives a construction of the localization C[W-1] as the quotient of C by the congruence relation determined by the homotopies on the sets of arrows C(X, Y). We develop here the 2-categorical localization, in which the 2-cells of this 2-localization are given by homotopies, and one can get the Quillen's localization when applying the connected components functor π0 on the hom-categories of the 2-localization. Our proof is not just a generalization of the well-known Quillen's one. We work with definitions of cylinders and homotopies introduced in [M.E. Descotte, E.J. Dubuc, M. Szyld; Model bicategories and their homotopy bicategories, arXiv:1805.07749 (2018)] considering only a single family of arrows Σ. When Σ is the class W of weak equivalences of a model category, we get the Quillen's results.