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 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 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.

Cigoli, A., & Metere, G. (2016). Extension theory and the calculus of butterflies. JOURNAL OF ALGEBRA, 458, 87-119 [10.1016/j.jalgebra.2016.03.015].

Extension theory and the calculus of butterflies

METERE, Giuseppe
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 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 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.
Settore MAT/02 - Algebra
Settore MAT/03 - Geometria
https://www.sciencedirect.com/science/article/pii/S0021869316300035
Cigoli, A., & Metere, G. (2016). Extension theory and the calculus of butterflies. JOURNAL OF ALGEBRA, 458, 87-119 [10.1016/j.jalgebra.2016.03.015].
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/10447/201442
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