Cauchy Completeness, Lax Epimorphisms and Effective Descent for Split Fibrations

For any suitable monoidal category V, we find that V-fully faithful lax epimorphisms in V V are precisely those V-functors F: A → B whose induced V-functors C F: C A → C B between the Cauchy completions are equivalences. For the case V= Set, this is equivalent to requiring that the induced functor F^*: CAT( A, Cat) → CAT( B, Cat) is an equivalence. By reducing the study of effective descent functors with respect to the indexed category of split (op)fibrations F to the study of the codescent factorization, we find that the observations above on fully faithful lax epimorphisms provide us with a characterization of (effective) F-descent morphisms in the category of small categories Cat; namely, we find that they are precisely the (effective) descent morphisms with respect to the indexed categories of discrete opfibrations,previously studied by Sobral. We include some comments on the Beck-Chevalley condition and future work.