Model bicategories and their homotopy bicategories

Abstract

We give the definitions of model bicategory and q-homotopy, which are natural generalizations of the notions of model category and homotopy to the context of bicategories. For any model bicategory C, denote by Cfc the full sub-bicategory of the fibrant-cofibrant objects. We prove that the 2-dimensional localization of C at the weak equivalences can be computed as a bicategory Ho(C) whose objects and arrows are those of Cfc and whose 2-cells are classes of q-homotopies up to an equivalence relation. When considered for a model category, q-homotopies coincide with the homotopies as considered by Quillen. The pseudofunctor C⟶qHo(C) which yields the localization is constructed by using a notion of fibrant-cofibrant replacement in this context. We include an appendix with a general result of independent interest on a transfer of structure for lax functors, that we apply to obtain a pseudofunctor structure for the fibrant-cofibrant replacement.