Facets of descent. II (Q1366309)

[For part I, see ibid. 2, No. 3, 245-281 (1994; Zbl 0805.18005).] The first question considered in this paper deals with the finding of these morphisms (in a category \({\mathcal C}\) with pullbacks) which are effective for descent with respect to every fibration over \({\mathcal C}\). The answer (obtained by using the methods of internal-category theory) is that the split epimorphisms are exactly the morphisms satisfying the above condition. Afterwards one shows how the notion of locally-split epimorphism can be generalized to the context of a (fairly) arbitrary indexed category \({\mathcal A}\) such that it still implies effective \({\mathcal A}\)-descent. Finally the composition and cancellation rules for effective descent morphisms are presented.