The 2-localization of a Quillen's model category

DUBUC, EDUARDO J.

Montreal

Seminario; McGill Seminar on Logic, Category Theory, and Computation; 2018

McGill University

We refer to Quillen's notion of a category C furnished with a model structure {W, F, coF} (weak equivalences, fibrations, cofibrations), and the construction of the localisation C[W−1] at W as the homotopy category Ho(C, W), that is, the quotient by the congruence determined by homotopies in the sets C(X, Y) of morphisms of C.In this talk I will construct the 2-localisation C[W−e] at W as the homotopy 2-category Ho~(C, W), a 2-category with the same arrows of C, and where the homotopies determine the 2-cells, instead of a congruence.A novel feature is the introduction of a generalisation of cylinder objects which allows the development of Quillen's theory of homotopies and the construction of the homotopy 2-category for an arbitrary category C and a single arbitrary class W ⊂ C. There is a 2-functor Ho~(C, W) → C[W−e], which is an isomorphism if W is split generated (an arrow f is split if there exists g such that fg = id or gf = id) and satisfies the 3 for 2 property.When W is the class of weak equivalences of a model category, and C is the category of fibrant-cofibrant objects, taking the set of connected components of the hom categories we obtain Quillen's results.This is a particular case of a joint work with Martin Szyld and Emilia Descotte, where we have developed a 2-dimensional version of a model category and its homotopy category. It corresponds to the case where the model bicategory is the trivial model bicategory determined by a model category.