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The aim of this article is to better understand the correspondence between$n$-cubic extensions and $3^n$-diagrams, which may be seen as non-abelianYoneda extensions, useful in (co)homology of non-abelian algebraic structures.We study a higher-dimensional version of the coequaliser/kernel pairadjunction, which relates $n$-fold reflexive graphs with $n$-fold arrows in anyexact Mal'tsev category. We first ask ourselves how this adjunction restrictsto an equivalence of categories. This leads to the concept of an effective$n$-fold equivalence relation, corresponding to the $n$-fold regularepimorphisms. We characterise those in terms of what (when $n=2$) Bourn callsparallelistic $n$-fold equivalence relations. We then further restrict theequivalence, with the aim of characterising the $n$-cubic extensions. We find acongruence distributivity condition, resulting in a denormalised $3^n$-Lemmavalid in exact Mal'tsev categories. We deduce a $3^n$-Lemma for short exactsequences in semi-abelian categories, which involves a distributivity conditionbetween joins and meets of normal subobjects. This turns out to be new even inthe abelian case.

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