6533b828fe1ef96bd12886a5
RESEARCH PRODUCT
Extension theory and the calculus of butterflies
Alan S. CigoliGiuseppe Meteresubject
TorsorCrossed moduleContext (language use)01 natural sciencesCohomologyCohomology; Extension; Fibrations; Obstruction theory; Schreier-mac lane theorem; TorsorsExtensionMathematics::Category Theory0103 physical sciences0101 mathematicsConnection (algebraic framework)MathematicsAlgebra and Number TheoryFunctorGroup (mathematics)010102 general mathematicsTorsorsExtension (predicate logic)Obstruction theorySchreier-mac lane theoremCohomologyFibrationsAlgebraSettore MAT/02 - AlgebraSchreier–Mac Lane theoremSettore MAT/03 - Geometria010307 mathematical physicsObstruction theorydescription
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–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.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2016-07-01 |