We first compare several algebraic notions of normality, from a\ud categorical viewpoint. Then we introduce an intrinsic description of\ud Higgins' commutator for ideal-determined categories, and we define a new\ud notion of normality in terms of this commutator. Our main result is to\ud extend to any semi-abelian category the following well-known\ud characterization of normal subgroups: a subobject K is normal in A if.\ud and only if, {[A, K] <= K. (C) 2010 Elsevier Inc. All rights reserved.

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

In this work we introduce the notions of Peiffer product and Peiffer commutator of internal pre-crossed modules over a fixed object B, extending the corresponding classical notions to any semi-abelian category C. We prove that, under mild additional assumptions on C, crossed modules are characterized as those pre-crossed modules X whose Peiffer commutator X, X is trivial. Furthermore we provide suitable conditions on C (fulfilled by a large class of algebraic varietes, including among others groups, associative algebras, Lie and Leibniz algebras) under which the Peiffer product realizes the coproduct in the category of crossed modules over B.

In a semi-abelian category, we give a categorical construction of the push forward of an internal pre-crossed module, generalizing the pushout of a short exact sequence in abelian categories. The main properties of the push forward are discussed. A simplified version is given for action accessible categories, providing examples in the categories of rings and Lie algebras. We show that push forwards can be used to obtain the crossed module version of the comprehensive factorization for internal groupoids. Indeed, for extensions with abelian kernel, the push forward construction makes the following classical assignment functorial: Opext C (Y, −) : [Abelian objects in C ↓ Y ] → [Abelian groups] (see [1] for groups and other algebraic categories, and [7] for pointed protomodular categories). Here, the supplementary conditions ensure that a gives rise to a morphism of abelian group objects in C ↓ Y. Indeed, the same situation can be treated also with the alternative approach developed by Bourn in [4], where a direction functor assigns to each abelian extension of the object Y a Y-module. Under this perspective, the push forward construction recovers the fact that such direction functor is a pseudo cofibration (see [4]). A generalization of the push forward construction arises when we consider the normal monomorphism appearing in any short exact sequence as an instance of a pre-crossed module. When the base category is the category of groups, under suitable hypothesis, it is possible to push forward along a map not only a normal monomorphism, but any pre-crossed module. This way we obtain a crossed module with the same cokernel (to the best of our knowledge, the push forward of a pre-crossed module was introduced by Noohi in [20]). The purpose of the present work is to develop a push forward construction in the intrinsic setting of a semi-abelian category C, where the notion of internal crossed module was introduced by Janelidze in [12]. In fact, in Theorem 3.6, we present necessary and sufficient conditions, expressed in terms of internal actions, for the push forward of a given pre-crossed module to exist in C (for the case of a crossed module, a push forward construction with different conditions was independently developed by Hartl [11]). These conditions simplify when the category C is action accessible [8], as presented in Theorem 3.10. The last result is very useful for the construction of push forward in many algebraic examples, like those of rings, Lie algebras, associative algebras and, more in general, any category of interest in the sense of Orzech [21]. The push forward construction, together with its main property (see Theorem 3.6, (PF)), turns out to be strongly related to the comprehensive factorization of internal functors (see [3]). Our investigation shows that push forwards can be used in order to factorize morphisms of crossed modules, so that final functors between internal groupoids can be characterized as push forward squares.

In this paper we start by pointing out that Yoneda's notion of a regular span S : X → A × B can be interpreted as a special kind of morphism, that we call fiberwise opfibration, in the 2-category Fib(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 P r0 : A × B → A is replaced by any split fibration over 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.