Resolutions, higher extensions and the relative Mal'tsev axiom

TL;DR

This paper explores the relationship between higher-dimensional extensions, simplicial resolutions, and the Mal'tsev property in regular categories, providing new insights and proofs connecting these concepts.

Contribution

It establishes a novel connection between higher extensions and simplicial resolutions, and offers a new proof characterizing Mal'tsev categories via the Kan property.

Findings

Resolutions correspond to higher extensions in all dimensions.

Stability conditions of extensions relate to the Kan property.

Regular categories are Mal'tsev iff all simplicial objects are Kan.

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.