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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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