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 H2-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–Mac Lane extension theorem [13] turns out to be an instance of our Theorem 6.3. Actually, even just in the case of groups, our approach reveals a result slightly more general than classical Schreier–Mac Lane theorem.
Extension theory and the calculus of butterflies / A.S. Cigoli, G. Metere. - In: JOURNAL OF ALGEBRA. - ISSN 0021-8693. - 458(2016 Jul 15), pp. 87-119. [10.1016/j.jalgebra.2016.03.015]
Extension theory and the calculus of butterflies
A.S. CigoliPrimo
;G. Metere
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 H2-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–Mac Lane extension theorem [13] turns out to be an instance of our Theorem 6.3. Actually, even just in the case of groups, our approach reveals a result slightly more general than classical Schreier–Mac Lane theorem.Pubblicazioni consigliate
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