Some recent categorical developments useful for universal algraists
by
Walter Tholen
Department of Mathematics and Statistics, York University, Toronto

The search by universal algebraists and categorists for suitable axioms which would describe the varieties of groups, rings, or of algebras as nicely as the abelian-category axioms capture the varieties of abelian groups or of modules is as old as their respective subjects. The first goal of my talk is to present a proposal (joint with G. Janelidze and L. Márki) for such axioms, based on D. Bourn's proto-modularity condition. Our notion of semi-abelian category is slightly stricter than that of a Maltsev category, but it lends itself perfectly to pursuing abstract group theory, general radical theory, and homology theory of non-abelian structures.

A second theme of general interest to both universal algebraists and categorists which I would like to explore further in this talk is that of injectivity. Reporting on joint work with J. Adámek, H. Herrlich, and J. Rosický, I wish to show how the notion of injective module or algebra, when extended naturally to homomorphisms, gives an immediate connection with Quillen's model categories. I shall present an existence criterion producing such structures in all varieties.

Date received: May 30, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caee-85.