Internal Categories and Groupoids in Congruence Modular Varieties Original Research Article

It is well known that there exist several simple descriptions of the internal categories and groupoids in various classical algebraic categories such as groups, rings, Lie algebras, etc. One of those descriptions is extended to arbitrary congruence modular varieties: we describe all internal categories satisfying a certain commutator condition which always holds for internal groupoids, and for all internal categories in Malʹtsev (= congruence permutable) varieties. The results of this paper show a deep connection between internal category theory and commutator theory in universal algebra; the so-called Kiss difference operation also plays a central role.