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Bicategories of fractions for groupoids in monadic categories. (English) Zbl 1350.18005
Summary: 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 $$\mathcal C$$, assuming that the axiom of choice holds in $$\mathcal C$$. For $$\mathbb T$$ a monad on $$\mathcal C$$, the bicategory of fractions of $$\mathrm{Grpd}(\mathcal C^{\mathbb{T}})$$ with respect to weak equivalences is now obtained replacing internal functors by what we call $$\mathbb T$$-monoidal functors. The notion of $$\mathbb T$$- monoidal functor is related to the notion of pseudo-morphism between strict algebras for a pseudo-monad on a 2-category.

##### MSC:
 18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories) 18C15 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads 18D05 Double categories, $$2$$-categories, bicategories and generalizations (MSC2010) 18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
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