Applied Categorical Structures 5: 229–248, 1997. 229 c  1997 Kluwer Academic Publishers. Printed in the Netherlands. Facets of Descent, II G. JANELIDZE Mathematical Institute of the Georgian Academy of Sciences, M. Alexidze Str. 1, 39003 Tbilisi, Georgia W. THOLEN Department of Mathematics and Statistics, York University, North York, Ontario, Canada M3J 1P3 (Received: 11 October 1994; accepted: 11 March 1996) Abstract. Methods of internal-category theory are applied to show that the split epimorphisms in a category C are exactly the morphisms which are effective for descent with respect to any fibration over C (or to any C-indexed category). In the same context, composition-cancellation rules for effective descent morphisms are established and being applied to (suitably defined) locally-split epimorphisms. Mathematics Subject Classifications (1991). 18D30, 18D35; 18A20, 18A25, 18B30. Key words: internal category, discrete cofibration, indexed category, (effective) descent morphism, (locally) split epimorphism, equivalence relation. Introduction As illustrated in this paper’s predecessor, finding sufficient (and necessary) con- ditions for a morphism in a given category to be effective for descent can be a challenging problem. In this paper, we concentrate on two particular facets of the descent problem and first consider the following question: 1. In any category C with pullbacks, which are the morphisms that are effective for descent with respect to every fibration E→C ? The short answer, given in Theorem 3.5 below, is that these are exactly the split epimorphisms of C . For its proof, we find it convenient to work with indexed categories, i.e., with pseudo-functors A: C op → CAT instead of fibrations E→C (cf. [12, 13]). Given such A, there is a canonical construction of an exponential A C for C being not merely an object of C but an internal category of C (cf. [10, Proposition A7]). Our Theorem 2.5 shows that this construction gives rise to a pseudo-functor of 2-categories A: cat(C ) op → CAT, a notion for which we give a precise definition in 2.4. As a consequence, one obtains that A preserves equivalences of categories, which is the main ingredient to our proof that split epimorphisms are effective A-descent morphisms. In case A is induced by the basic fibration, already in [8] we had given a direct proof of the latter result. Together with the observation that in the category Top