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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 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.

We give necessary and sufficient conditions on a functor k : C -> epsilon, where C is an algebraic theory, in for the induced functor epsilon(k-,-) : epsilon -> Alg(C) to be a geometric morphism or a localization. We apply our techniques also to the particular case of module categories and to the case of presheaf categories. (c) 2006 Elsevier Inc. All rights reserved.

A categorical group is a monoidal groupoid in which each object has a tensorial inverse. Two main examples are the Picard categorical group of a monoidal category and the Brauer categorical group of a braided monoidal category with stable coequalizers. After discussing the notions of kernel, cokemel and exact sequence for categorical groups, we show that, given a suitable monoidal functor between two symmetric monoidal categories with stable coequalizers, it is possible to build up a five-term Picard-Brauer exact sequence of categorical groups. The usual Units-Picard and Picard-Brauer exact sequences of abelian groups follow from this exact sequence of categorical groups. We also discuss the direct sum decomposition of the Brauer-Long group. (C) 2002 Elsevier Science B.V. All rights reserved.

Using the exact completion of a weakly left exact category, we specialize previous results on monadic categories over LET to obtain a characterization of localizations of finitary algebraic categories. As a corollary, we have the Popescu-Gabriel representation theorem for Grothendieck categories. (C) 1998 Elsevier Science B.V. All rights reserved.

The regular and exact completions of categories with weak limits are proved to exist and to be determined by an appropriate universal property. Several examples are discussed, and in particular the class of examples given by categories monadic over a power of Set: any such a category is in fact the exact completion of the full subcategory of free algebras. Applications to Grothendieck toposes and geometric morphisms, and to epireflective hulls are also discussed. (C) 1998 Elsevier Science B.V.

We show that the cartesian closed category of compactly generated Hausdorff spaces is regular, but is neither exact, nor locally cartesian closed. In fact we find a coequalizer of an equivalence relation which is not stable under pullback.

It is known that monoidal functors between internal groupoids in the category Grp of groups constitute the bicategory of fractions of the 2-category Grpd(Grp) of internal groupoids, internal functors and internal natural transformations in Grp, with respect to weak equivalences (that is, internal functors which are internally fully faithful and essentially surjective on objects). Monoidal functors can be equivalently described by a kind of weak morphisms introduced by B. Noohi under the name of butterflies. In order to internalize monoidal functors in a wide context, we introduce the notion of internal butterflies between internal crossed modules in a semi-abelian category C, and we show that they are morphisms of a bicategory B(C). Our main result states that, when in C the notions of Huq commutator and Smith commutator coincide, then the bicategory B(C) of internal butterflies is the bicategory of fractions of Grpd(C) with respect to weak equivalences. (C) 2013 Elsevier Inc. All rights reserved.

We consider various (free) completion processes: the exact completion and the regular completion of a category with weak finite limits, the pre-regular completion of a category with finite products and weak finite limits, the exact completion of a regular category, the regular reflection of a pre-regular category, and the filtered-colimit completion of a small category. In each case we give necessary and sufficient conditions for the completion to be extensive; or, in the case of the pre-regular completion, for the completion to satisfy a weakened notion of extensivity which we call pre-extensivity. (C) 2001 Elsevier Science B.V. All rights reserved.