Some insights on bicategories of fractions – I

In this paper we investigate the construction of bicategories of fractions originally described by D. Pronk: given any bicategory C together with a suitable class of morphisms W, one can construct a bicategory C[W^{-1}], where all the morphisms of W are turned into internal equivalences, and that is universal with respect to this property. Most of the descriptions leading to such a construction were long and heavily based on the axiom of choice. In this paper we simplify considerably the constructions of associators, vertical and horizontal compositions in a bicategory of fractions, thus proving that the axiom of choice is not needed under certain conditions. The simplified description of associators and 2-compositions will also play a crucial role in the next papers of this series.