Resolutions, higher extensions and the relative Mal'tsev axiom

Abstract

We study how the concept of higher-dimensional extension which comes from categorical Galois theory relates to simplicial resolutions. For instance, an augmented simplicial object is a resolution if and only if its truncation in every dimension gives a higher extension, in which sense resolutions are infinite-dimensional extensions or higher extensions are finite-dimensional resolutions. We also relate certain stability conditions of extensions to the Kan property for simplicial objects. This gives a new proof of the fact that a regular category is Mal'tsev if and only if every simplicial object is Kan, using a relative setting of extensions.

Dedicated to James Gray and Tamar Janelidze on the occasion of their wedding. 23 pages. Fourth section from previous version removed, will become a separate article

References in corpus (8)

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• On the second cohomology group in semi-abelian categories
• Higher central extensions and Hopf formulae
• Protoadditive functors, derived torsion theories and homology
• Relative Commutator Theory in Semi-Abelian Categories
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Cited by in corpus (11)

• Higher central extensions and cohomology
• The Yoneda isomorphism commutes with homology
• Higher Central Extensions in Mal'tsev Categories
• The Cuboid Lemma and Mal'tsev categories
• Higher commutator conditions for extensions in Mal'tsev categories
• On the "three subobjects lemma" and its higher-order generalisations
• A comonadic interpretation of Baues-Ellis homology of crossed modules
• Higher coverings of racks and quandles -- Part I
• A symmetric approach to higher coverings in categorical Galois theory
• Non-additive derived functors via chain resolutions
• Fundamental groupoids for simplicial objects in Mal'tsev categories