Effective descent morphisms in some quasivarieties of algebraic, relational, and more general structures
by
Ana Helena Roque
Universidade de Aveiro, Portugal

In a quasivariety of models of a first order language, regular epimorphisms are the same as strong surjective homomorphisms, and closeness under coequalizers is equivalent to closeness under strong homomorphic images. Moreover, we show that the class of effective descent morphisms in any quasivariety closed under strong homomorphic images coincides with the class of regular epimorphisms-yielding many new concrete examples of non-exact regular categories, in which every regular epimorphism is an effective descent morphism.

We also extend these arguments to a more general setting, where:

• the category of sets is replaced by an abstract exact category S, whose objects hence play the role of underlying sets of the structures we study;
• instead of specifying a first order language, we specify two sets F and R of endofunctors of S, in which each F in F preserves regular epimorphisms, and each R in R preserves regular epimorphisms and finite limits;
• a model (or a structure) for the data above is a system A = (A, (F^A)_{F in F}, (R^A)_{R in R}), in which A is an object in S, F^A a morphism from F(A) to A, and R^A a subobject in A.

We then analyse further exactness properties of these structures, in connection with the theory of generalized central extensions.

This work is a part of the author's Ph.D. thesis; supervisor Prof. G. Janelidze.

Date received: May 20, 2002