Model bicategories and their homotopy bicategories

DESCOTTE, M. EMILIA; DUBUC, EDUARDO J; SZYLD, MARTIN

ADVANCES IN MATHEMATICS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 2022

0001-8708

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 $mathcal{C}$, 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 $mathcal{H}{{o}}(mathcal{C})$ whose objects and arrows are those of $mathcal{C}_{fc}$ 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 $mathcal{C} stackrel{q}{longrightarrow} mathcal{H}{{o}}(mathcal{C})$ 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.}