Higher extensions in exact Mal'tsev categories: distributivity of congruences and the $3^n$-Lemma

Abstract

The aim of this article is to better understand the correspondence between n-cubic extensions and 3 n -diagrams, which may be seen as nonabelian Yoneda extensions, useful in (co)homology of non-abelian algebraic structures.We study a higher-dimensional version of the coequaliser/kernel pair adjunction, which relates n-fold reflexive graphs with n-fold arrows in any exact Mal'tsev category.We first ask ourselves how this adjunction restricts to an equivalence of categories. This leads to the concept of an effective n-fold equivalence relation, corresponding to the n-fold regular epimorphisms. We characterise those in terms of what (when n " 2) Bourn calls parallelistic n-fold equivalence relations.We then further restrict the equivalence, with the aim of characterising the n-cubic extensions. We find a congruence distributivity condition, resulting in a denormalised 3 n -Lemma valid in exact Mal'tsev categories. We deduce a 3 n -Lemma for short exact sequences in semi-abelian categories, which involves a distributivity condition between joins and meets of normal subobjects. This turns out to be new even in the abelian case.