Extension theory and the calculus of butterflies

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• 作者：Alan S. Cigoli^a ; ^{alan.cigoli@unimi.it" class="auth_mail" title="E-mail the corresponding author} ; Giuseppe Metere^b
• 关键词：18G50 ; 08C05 ; 20J06 ; 18D30
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 July 2016
• 年：2016
• 卷：458
• 期：Complete
• 页码：87-119
• 全文大小：590 K

文摘

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$H^2$-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–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.