A categorical approach to commutator theory
This article continues a theme the author, together with A. Carboni and others, has been pursuing since the beginning of the decade: the categorical analysis of Mal'tsev's characterization of those varieties in which every reflexive relation is already a congruence [cf. \textit{A. Carboni}, \textit{J. Lambek} and the author, J. Pure Appl. Algebra 69, 271-284 (1990; Zbl 0722.18005); \textit{A. Carboni}, \textit{G. M. Kelly} and the author, Appl. Categ. Struct. 1, 385-421 (1993; Zbl 0799.18002); \textit{A. Carboni}, the author and \textit{N. Pirovano}, Proc. Int. Summer Category Theory Meeting, Montreal 1991, CMS Conf. Proc. 13, 97-109 (1992; Zbl 0791.18005)]. The categorical approach takes the notion of a well-behaved lattice of equivalence relations for each object as primitive. Here this leads to the consideration of exact Mal'tsev categories. For these a notion of commutator is defined. The main result concerns the characterization of this commutator by means of a universal property. [Reviewer's remark: The references contain a surprising number of errors].
well-behaved lattice of equivalence relations
scientific article; zbMATH DE number 827369