Search results for "crossed module"
showing 5 items of 5 documents
INTERNAL CROSSED MODULES AND PEIFFER CONDITION
2010
In this paper we show that in a homological category in the sense of F. Borceux and D. Bourn, the notion of an internal precrossed module corresponding to a star-multiplicative graph, in the sense of G. Janelidze, can be obtained by directly internalizing the usual axioms of a crossed module, via equivariance. We then exhibit some sufficient conditions on a homological category under which this notion coincides with the notion of an internal crossed module due to G. Janelidze. We show that this is the case for any category of distributive Omega(2)-groups, in particular for the categories of groups with operations in the sense of G. Orzech.
Extension theory and the calculus of butterflies
2016
Abstract This paper provides a unified treatment of two distinct viewpoints concerning the classification of group extensions: the first uses weak monoidal functors, the second classifies extensions by means of suitable H 2 -actions. We develop our theory formally, by making explicit a connection between (non-abelian) G-torsors and fibrations. Then we apply our general framework to the classification of extensions in a semi-abelian context, by means of butterflies [1] between internal crossed modules. As a main result, we get an internal version of Dedecker's theorem on the classification of extensions of a group by a crossed module. In the semi-abelian context, Bourn's intrinsic Schreier–M…
External derivations of internal groupoids
2008
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.
Peiffer product and peiffer commutator for internal pre-crossed modules
2017
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 varieties, 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.
A Push Forward Construction and the Comprehensive Factorization for Internal Crossed Modules
2014
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.