  Given a bicategory C and a family W of arrows of C, we give conditions on the
pair (C,W) that allow us to construct the bicategorical localization with
respect to W by dealing only with the 2-cells, that is without adding objects
or arrows to C. We show that in this case, the 2-cells of the localization can
be given by the homotopies with respect to W, a notion defined in this article
which is closely related to Quillen's notion of homotopy for model categories
but depends only on a single family of arrows. This localization result has a
natural application to the construction of the homotopy bicategory of a model
bicategory, which we develop elsewhere, as the pair (C_{fc},W) given by the
weak equivalences between fibrant-cofibrant objects satisfies the conditions
given in the present article.
