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. Cigoli
Primo
;
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.
cohomology; extension; fibrations; obstruction theory; Schreier-Mac Lane theorem; torsors
Settore MAT/02 - Algebra
15-lug-2016
8-apr-2016
Article (author)
File in questo prodotto:
Non ci sono file associati a questo prodotto.
Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/378047
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 9
  • ???jsp.display-item.citation.isi??? 8
social impact