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 T-monoidal functor is related to the notion of pseudo-morphism between strict algebras for a pseudo-monad on a 2-category.

Tbilisi Mathematical Journal, Vol. 8(1) (2015), pp. 85-105