INTERNAL CATEGORIES AND GROUPOIDS IN CONGRUENCE MODULAR VARIETIES

It is well known that there exist several simple descriptions of the i nternal categories and groupoids in various classical algebraic catego ries such as groups, rings, Lie algebras, etc. One of those descriptio ns is extended to arbitrary congruence modular varieties: we describe all internal categories satisfying a certain commutator condition whic h always holds for internal groupoids, and for all internal categories in Mal'tsev (= congruence permutable) varieties. The results of this paper show a deep connection between internal category theory and comm utator theory in universal algebra; the so-called Kiss difference oper ation also plays a central role. (C) 1997 Academic Press.