Internal Categories and Groupoids in Congruence Modular Varieties

✍ Scribed by George Janelidze; Maria Cristina Pedicchio

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 221 KB
• Volume: 193
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1996.6970

✦ Synopsis

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 Ž . internal categories in Mal'tsev scongruence 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. ᮊ 1997 Academic Press N Ž . w x ii Ker s, Ker b s 1 Ž . where 1 denotes a trivial group . 552