Effective descent morphisms in some quasivarieties of algebraic, relational, and more general structures (Q1771122)

Let us concern the general motivation (from the reviewer's point of view) for this paper first, leaving its results out for a while. The existence of effective descent morphisms can be considered naturally when the question of monadicity (also called tripleability) is being raised, that is the question of deciding whether a functor \(U: \mathcal{B}\to \mathcal{C}\) with a left adjoint has the property that the Eilenberg-Moore category for the corresponding monad is essentially the same as \(\mathcal{B}\). If the Eilenberg-Moore comparison functor \(\Phi\) for \(U\) is full and faithful, \(U\) is said to be of descent type (or a descent morphism); and if \(\Phi\) is an equivalence of categories, it is said to be of effective descent type (or an effective descent morphism). It is known that every variety of universal algebras is an exaxt category (i.e., it is regular and every equivalence relation is a kernel pair of some arrow). This is not the case for quasivarieties (that are the categories which are just regular). In the paper, it is proved that the class of effective descent morphisms coincides with the class of regular epimorphisms in any prevariety of first-order structures. This had been done on a general level of an exact category of suitable internal structures introduced in the paper in a somewhat Mal'cevian way.