Bicategories of fractions for groupoids in monadic categories

Abstract

The bicategory of fractions of the 2-category of internal groupoids and internal functors in groups with respect to weak equivalences (i.e., functors which are internally full, faithful and essentially surjective) has an easy description: one has just to replace internal functors by monoidal functors. In the present paper, we generalize this result from groups to any monadic category over a regular category C; assuming that the axiom of choice holds in C: For T a monad on C; the bicategory of fractions of Grpd(C T ) with respect to weak equivalences is now obtained replacing internal functors by what we call T-monoidal functors. The notion of Tmonoidal functor is related to the notion of pseudo-morphism between strict algebras for a pseudo-monad on a 2-category.