A (somewhat) biased look at dominions and epimorphisms
by
Arturo Magidin
Instituto de Matem/'aticas, Universidad Nacional Autonoma de Mexico

An epimorphism is a right-cancellable morphism. In many familiar categories a morphism is epic if and only if it is surjective; however, in others (for example, Semigroups, Torsion-Free Abelian Groups) there are epimorphisms which are not surjective. The guiding question is: In a given category, are all epis surjective? If not, can we describe in an 'effective' way the epimorphisms? Although it may seem an artificial question, it has led to a lot of very interesting mathematics.

Among these are the dominions of Isbell, which provide a tool for investigating epimorphisms, as well as a connection to the theory of amalgams. I will take a look at some results, some recent and some not, on the subject of dominions and epimorphisms in various categories. I will pay particular attention to areas which have not received as much attention in the literature (such as distributive lattices and varieties of groups); this will make the look on the subject somewhat biased. In particular, I will talk about the result (due to David Wasserman at Berkeley) that in the category of all distributive lattices, dominions are stable, and a result which reduces the problem of describing epimorphisms in varieties of groups to the indecomposable varieties.

Date received: May 18, 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 # cafg-06.