Resolutions, higher extensions and the relative Mal'tsev axiom
  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. 
Tomas Everaert, Julia Goedecke and Tim Van der Linden
math.CT