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.
Categories: math.CT