Anticommutativity and the triangular lemma

Abstract

For a variety $\mathcal{V}$, it has been recently shown that binary products commute with arbitrary coequalizers locally, i.e., in every fibre of the fibration of points $\pi: \mathrm{Pt} (\mathbb{C}) \rightarrow \mathbb{C}$, if and only if Gumm's shifting lemma holds on pullbacks in $\mathcal{V}$. In this paper, we establish a similar result connecting the so-called triangular lemma in universal algebra with a certain categorical $\textit{anticommutativity}$ condition. In particular, we show that this anticommutativity and its local version are Mal'tsev conditions, the local version being equivalent to the triangular lemma on pullbacks. As a corollary, every locally anticommutative variety $\mathcal{V}$ has directly decomposable congruence classes in the sense of Duda, and the converse holds if $\mathcal{V}$ is idempotent.