Categories as Algebra II
by
Benjamin Steinberg
University of Porto
Coauthors: Bret Tilson

This talk surveys some recent results by Tilson and the author in using categories to understand the semidirect product operator on monoids and pseudovarieties of such. In the second author's paper from 1987, he proved the derived category theorem which characterizes membership in a pseudovariety V*W in terms of membership in W and category membership in V. This paper is motivated by interest in semidirect products of more than two pseudovarieties. For this, it is necessary to introduce notions of a semidirect product of categories and a derived category for relational morphisms of categories. These constructions stand in an adjoint-like relation (that is, the derived category has a certain universal property with respect to the semidirect product).

Our primary result is that if f is a relational morphism of categories, then D_f belongs to V*W if and only if f = gh with D_gin V and D_h in W. From this result, we can deduce the associativity of the semidirect product operator from the associativity of relational morphism composition. We can also show that category membership in monoid pseudovarieties commutes with the semidirect product (that is, the global operator commutes with the semidirect product).

The usual description of free objects over semidirect products can also be shown to be a consequence of our work.

If time permits, I will also indicate some related results on inverse semigroups obtained by using the above techniques on inverse semigroups viewed as ordered groupoids.

http://www.fc.up.pt/cmup/home/bsteinbg

Date received: April 28, 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 # caec-13.