Resolutions, higher extensions and the relative Mal'tsev axiom
arXiv:1206.2946 · doi:10.1016/j.jalgebra.2012.07.036
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