Preprint Number 10–49 RESOLUTIONS, HIGHER EXTENSIONS AND THE RELATIVE MAL’TSEV AXIOM

Abstract: We study how the concept of higher-dimensional extension which co-mes 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 di-mension 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