Quillen's Model Categories revisited

DUBUC, EDUARDO J.

Ponta Delgada

Congreso; CT2018; 2018

University of Azores

We refer to Quillen's notion of a category C furnished with a model structurefW; F; coFg (weak equivalences, brations, cobrations), and the construction of thelocalization of the class W as the quotient by the congruence determined by homotopiesin the sets C(X; Y ) of morphisms of C [Homotopical Algebra, Springer LNM 43]. Injoint (unpublished) work with Martin Szyld and Emilia Descotte we have developed a2-dimensional generalization of this notion and construction.In this talk I will consider the 2-dimensional theory in the particular case in which themodel bicategory is the trivial model bicategory determined by a model category. In thiscase the computations are considerable simpler, and our theory yields new results in theordinary theory of localization of categories.The novel feature is the introduction of a generalization of cylinder objects which allowsthe construction of the homotopy congruence associated to an arbitrary single class , andits quotient homotopy category Ho(C; ). There is a functor Ho(C; ) 􀀀! C[􀀀1], whichis an isomorphism if is split generated (an arrow f is split if there exists g such thatfg = id or gf = id) and satises the 3 for 2 property. When is the class W of a modelcategory, we obtain Quillen's results.1