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Extensions with Abelian Kernels in Protomodular Categories
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D. Bourn
Published/Copyright:
March 3, 2010
Abstract
As observed by J. Beck, and as we know from M. Barr's and his joint work on triple cohomology, the classical isomorphism Opext ≅ 𝐻2 that describes group extensions with abelian kernels, can be deduced from the equivalence between such extensions and torsors (in an appropriate sense). The same is known for many other “group-like” algebraic structures, and now we present a purely-categorical version of that equivalence, essentially by showing that all torsors are extensions with abelian kernels in any pointed protomodular category, and by giving a necessary and sufficient condition for the converse.
Key words and phrases:: Protomodular category; semi-abelian category; Opext functor; extension; torsor
Received: 2004-06-10
Published Online: 2010-03-03
Published in Print: 2004-December
© Heldermann Verlag
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Articles in the same Issue
- Complex Points of Two-Dimensional Surfaces
- On Homology and Cohomology Groups of Remainders
- Strong Homology Theories as Localizations
- Extensions with Abelian Kernels in Protomodular Categories
- Polytopes and 𝐾-Theory
- Mappings Confluent over Locally Connected Continua
- Equicontinuity and Quasi-Uniformities
- Witt's Theorem for Groups with Action and Free Leibniz Algebras
- Connectedness of a Fine Topology and Localization in Bitopological Spaces
- Minimal Free Multi-Models for Chain Algebras
- On Generalized Step-Functions and Superposition Operators
- New Applications of Algebraic Formulae for Topological Invariants
- On the Shape of Regularly Movable Compacta
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- More on Descent Theory for Schemes
Keywords for this article
Protomodular category;
semi-abelian category;
Opext functor;
extension;
torsor
Articles in the same Issue
- Complex Points of Two-Dimensional Surfaces
- On Homology and Cohomology Groups of Remainders
- Strong Homology Theories as Localizations
- Extensions with Abelian Kernels in Protomodular Categories
- Polytopes and 𝐾-Theory
- Mappings Confluent over Locally Connected Continua
- Equicontinuity and Quasi-Uniformities
- Witt's Theorem for Groups with Action and Free Leibniz Algebras
- Connectedness of a Fine Topology and Localization in Bitopological Spaces
- Minimal Free Multi-Models for Chain Algebras
- On Generalized Step-Functions and Superposition Operators
- New Applications of Algebraic Formulae for Topological Invariants
- On the Shape of Regularly Movable Compacta
- Generalized Heisenberg Groups and Shtern's Question
- More on Descent Theory for Schemes
