  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.
