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We set up a fibred categorical theory of obstruction and classification of morphisms that specialises to the one of monoidal functors between categorical groups and also to the Schreier-Mac Lane theory of group extensions. Further applications are provided to crossed extensions and crossed bimodule butterflies, with in particular a classification of non-abelian extensions of unital associative algebras in terms of Hochschild cohomology.

We characterize fibrations and $$*$$ -fibrations in the 2-category of internal groupoids in terms of the comparison functor from certain pullbacks to the corresponding strong homotopy pullbacks. As an application, we deduce the internal version of the Brown exact sequence for $$*$$ -fibrations from the internal version of the Gabriel–Zisman exact sequence. We also analyse fibrations and $$*$$ -fibrations in the category of arrows and study when the normalization functor preserves and reflects them. This analysis allows us to give a characterization of protomodular categories using strong homotopy kernels and a generalization of the Snake Lemma.

In this paper we explore the snail lemma in a pointed regular category. In particular, we show that under the presence of cokernels of kernels, the validity of the snail lemma is equivalent to subtractivity of the category. As a corollary, this gives that in the more restrictive context of a normal category the validity of the snail lemma is equivalent to the validity of the snake lemma.

In this paper we start by pointing out that Yoneda's notion of a regular span $S \colon \mathcal{X} \to \mathcal{A} \times \mathcal{B}$ can be interpreted as a special kind of morphism, that we call fiberwise opfibration, in the 2-category $\mathsf{Fib}(\mathcal{A})$. We study the relationship between these notions and those of internal opfibration and two-sided fibration. This fibrational point of view makes it possible to interpret Yoneda's Classification Theorem given in his 1960 paper as the result of a canonical factorization, and to extend it to a non-symmetric situation, where the fibration given by the product projection $Pr_0 \colon \mathcal{A} \times \mathcal{B} \to \mathcal{A}$ is replaced by any split fibration over $\mathcal{A}$. This new setting allows us to transfer Yoneda's theory of extensions to the non-additive analog given by crossed extensions for the cases of groups and other algebraic structures.

We establish conditions giving the existence of bipullbacks in bicategories of fractions. We apply our results to construct a π 0 - π 1 exact sequence associated with a fractor between groupoids internal to a pointed exact category.

We establish a generalized form both of the Gabriel-Zisman exact sequence associated with a pointed functor between pointed groupoids, and of the Brown exact sequence associated with a fibration of pointed groupoids. Our generalization consists in replacing pointed groupoids with groupoids internal to a pointed regular category with reflexive coequalizers.

We study internal profunctors and their normalization under various conditions on the base category. In the Mal'tsev case we give an easy characterization of profunctors. Moreover, when the base category is regular with any regular epimorphism effective for descent, we characterize those profunctors which are fractions of internal functors with respect to weak equivalences. (C) 2012 Elsevier B.V. All rights reserved.

If His a G-crossed module, the set of derivations of Gin H is a monoid under the Whitehead product of derivations. We interpret the Whitehead product using the correspondence between crossed modules and internal groupoids in the category of groups. Working in the general context of internal groupoids in a finitely complete category, we relate derivations to holomorphisms, translations, affine transformations, and to the embedding category of a groupoid. (C) 2007 Elsevier B.V. All rights reserved.

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.

We define the cohomology categorical groups of a complex of symmetric categorical groups, and we construct a long 2-exact sequence from an extension of complexes. As special cases, we obtain Ulbrich cohomology of Picard categories and the Hattori-Villamayor-Zelinsky sequences associated with a ring homomorphism. Applications to simplicial cohomology with coefficients in a symmetric categorical group, and to derivations of categorical groups are also discussed. (C) 2004 Elsevier B.V. All rights reserved.

Starting from known examples of factorization systems in 2-categories, we discuss possible definitions of proper factorization system in a 2-category. We focus our attention on the construction of the free proper factorization system on a given 2-category. (C) 2003 Elsevier Science B.V. 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.

We associate to a continuous map between pointed spaces a long 2-exact sequence of homotopy pointed groupoids. The usual homotopy sequence of a map follows from this 2-exact sequence taking, for each groupoid, the set of connected components. We also study a condition of strong 2-exactness for a sequence of cat-groups and pointed groupoids.

This paper is an attempt to study extensions of symmetric categorical groups from a structural point of view. Using in a systematic way bilimits in the 2-category of symmetric categorical groups, we develop a theory which closely follows the classical theory of abelian group extensions. The basic results are established for any proper class of extensions, and a cohomological classification is obtained for those extensions whose epi part has a categorical section.

We show how the exact completion of a regular category constitutes a unifying framework for the abstract characterization of various classes of quasi-varieties.

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.

We discuss the notion of bimodel in order to obtain a classification of the equivalences between categories of models in the sense of functorial semantics.

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.

To determine the effects of anabolic steroids on serum lipid and apoprotein levels, 14 white male body builders who self-administered steroids for 2 to 3 months (steroid users) were studied; 10 agreed to screening while they were taking the drugs (ON treatment) and also at about 3 months following their suspension (OFF treatment). Controls consisted of 17 body builders who had never taken steroids (nonusers), and a group of 18 healthy sedentary subjects (controls). During the period of steroid administration, there was a slight reduction in total serum cholesterol, with a marked cholesterol decrease in the high-density lipoprotein (HDL) subfractions HDL2 and HDL3, and a significant reduction in the HDL2 cholesterol/HDL3 cholesterol ratio; the percentage of serum cholesterol transported by low-density lipoproteins (LDL) increased significantly. In addition, a marked apoprotein (apo) A-I reduction in the HDL2 and HDL3 subfractions was observed, as well as an apo A-II decrease that was significant only in the HDL3 subfraction, with an A-I/A-II ratio significantly reduced in both subfractions. Serum apo B was only slightly increased, with a very high B/A-I ratio. Apolipoprotein C-II and E levels showed no modifications, while apo C-III reduced significantly. Lipid and apoprotein values returned to almost normal levels in the OFF treatment period. Findings in the group of nonusers were similar to those in sedentary subjects. These results indicate that anabolic steroids profoundly alter the serum lipid-protein profile, and the changes may be caused in part by the significant differences observed in apoprotein levels.

Working in the context of categorical groups, we show that the semidirect product provides a biequivalence between actions and points. From this biequivalence, we deduce a two-dimensional classification of split extensions of categorical groups, as well as the universal property of the holomorph of a categorical group. We also discuss the link between the holomorph and inner autoequivalences.

When the exact completion of a category with weak finite limits is a Mal’cev category, it is possible to combine the universal property of the exact completion and the universal property of the coequalizer completion. We use this fact to explain Freyd’s representation theorems in abelian and Frobenius categories.