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 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.