Variations of the Shifting Lemma and Goursat categories

TL;DR

This paper characterizes Mal'tsev and Goursat categories using stronger forms of the Shifting Lemma, providing new insights into their structure through relations in regular categories.

Contribution

It introduces novel characterizations of Mal'tsev and Goursat categories via the Shifting Lemma applied to specific types of relations, extending classical congruence-based descriptions.

Findings

Mal'tsev categories characterized by the Shifting Lemma for reflexive relations.

Goursat categories characterized by the Shifting Lemma for reflexive and positive relations.

Provides new characterizations of 2- and 3-permutable varieties.

Abstract

We prove that Mal'tsev and Goursat categories may be characterised through stronger variations of the Shifting Lemma, that is classically expressed in terms of three congruences $R$, $S$ and $T$, and characterises congruence modular varieties. We first show that a regular category $\mathcal C$ is a Mal'tsev category if and only if the Shifting Lemma holds for reflexive relations on the same object in $\mathcal C$. Moreover, we prove that a regular category $\mathcal C$ is a Goursat category if and only if the Shifting Lemma holds for a reflexive relation $S$ and reflexive and positive relations $R$ and $T$ in $\mathcal C$. In particular this provides a new characterisation of $2$-permutable and $3$-permutable varieties and quasi-varieties of universal algebras.