Model bicategories and their homotopy bicategories

We give the definitions of model bicategory and $w$-homotopy, which are natural generalizations of the notions of model category and homotopy to the context of bicategories. For any model bicategory $\mathcal{C}$, denote by $\mathcal{C}_{fc}$ the full sub-bicategory of the fibrant-cofibrant objects. We prove that the 2-dimensional localization of $\mathcal{C}$ at the weak equivalences can be computed as a bicategory $\mathcal{H}\mbox{o}(\mathcal{C})$ whose objects and arrows are those of $\mathcal{C}_{fc}$ and whose 2-cells are classes of $w$-homotopies up to an equivalence relation. The pseudofunctor $\mathcal{C} \stackrel{r}{\longrightarrow} \mathcal{H}\mbox{o}(\mathcal{C})$ which yields the localization is constructed by using a notion of fibrant-cofibrant replacement in this context. When considered for a model category, the results in this article give in particular a bicategory whose reflection into categories is the usual homotopy category constructed by Quillen.