1 Nov 2005 2:30 - 4:00 Claudio Hermida Descent on 2-fibrations and strong 2-regularity(*) Abstract: We consider pseudo-descent in the context of 2-fibrations. A 2-category of descent data is associated to a 3-truncated simplicial object in the base 2-category. A morphism q in the base induces (via comma-objects and pullbacks) an internal category whose truncated nerve allows the definition of the 2-category of descent data for q . When the 2-fibration admits direct images, we provide the analogous of the Beck-Benabou-Roubaud theorem, identifying the 2-category of descent data with that of pseudo-algebras for the pseudo-monad q*N#q . We introduce a notion of 2-regularity for a 2-category R, so that its basic 2-fibration of internal fibrations cod: Fib (R) b^F^R R admits direct images. In this context, we show that essentially-surjective-on-objects morphisms, defined by a certain lax colimit, are of effective descent by means of a Beck-style pseudo-monadicity theorem. (*) in Applied Categorical Structures, 12(5-6), 427--459, 2004.