The Cuboid Lemma and Mal'tsev categories

TL;DR

This paper characterizes Mal'tsev categories via a new diagrammatic condition called the Cuboid Lemma, extending the understanding of categorical properties through a generalized $3 imes 3$ Lemma involving pullbacks of regular epimorphisms.

Contribution

It introduces the Cuboid Lemma as a new characterization of Mal'tsev categories, generalizing previous results and extending to relative contexts.

Findings

The Cuboid Lemma characterizes Mal'tsev categories.

The lemma involves pullbacks of regular epimorphisms.

Extension of results to relative settings.

Abstract

We prove that a regular category $\mathcal C$ is a Mal'tsev category if and only if a strong form of the denormalised $3 \times 3$ Lemma holds true in $\mathcal C$. In this version of the $3 \times 3$ Lemma, the vertical exact forks are replaced by pullbacks of regular epimorphisms along arbitrary morphisms. The shape of the diagram it determines suggests to call it the Cuboid Lemma. This new characterisation of regular categories that are Mal'tsev categories (= $2$-permutable) is similar to the one previously obtained for Goursat categories (= $3$-permutable). We also analyse the "relative" version of the Cuboid Lemma and extend our results to that context.