An overview of ternary commutators: Bulatov, Higgins, Smith

The aim of this talk is to explain how a three-dimensional version of Pedicchio's categorical approach to the binary Smith commutator of Universal Algebra allows us to compare three ternary commutators which occur naturally in mathematics but are a priori very different: the ternary Bulatov commutator~\cite{Bulatov}, a universal-algebraic concept meant to generalise the binary commutator of two congruences to a commutator of three congruences; the ternary Higgins commutator, first introduced in the context of varieties of $\Omega$-groups, of which recently a categorical version was developed; the ternary Smith commutator, which is supposed to allow an interpretation of cohomology via higher extensions as in beyond the case of trivial coefficients = higher central extensions. Our first aim is to capture the Bulatov commutator through a definition which is valid in exact Mal'tsev categories. We then show that in the context of an algebraically coherent semi-abelian category, the ternary Higgins commutator of a triple of normal subobjects corresponds to the normalisation of the ternary Bulatov commutator of the corresponding equivalence relations. This gives a first, indirect answer to the question, in which sense the ternary Higgins commutator expresses that ``three things commute''. On the way, we sketch a general result on commuting triples of arrows, crucial here and in our proposed definition of a ternary Smith commutator. We then explain how this is related to the approach to cohomology in my work with Diana Rodelo. This talk is based on joint work with her and with Cyrille Simeu.