On the ternary commutator, I: Exact Mal'tsev categories

Cyrille Sandry Simeu and Tim Van der Linden

In this first article on the Bulatov commutator, we introduce a ternary commutator of equivalence relations in the context of an exact Mal'tsev category with binary coproducts. We prove that, for Mal'tsev varieties, our notion is a particular case (where n=3) of the n-fold commutator introduced (originally in the context of Mal'tsev algebras) by A. Bulatov. We study its basic stability properties as well as the relationship with the (binary) Smith-Pedicchio commutator. In a forthcoming second article, we restrict the context to algebraically coherent semi-abelian categories, where we prove that the commutator introduced here corresponds to the ternary Higgins commutator of M. Hartl and the second author.

Keywords: Higher commutator; exact Mal'tsev category

2020 MSC: 18E13

Theory and Applications of Categories, Vol. 36, 2021, No. 15, pp 379-422.

Published 2021-06-23.


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