Cauchy Completeness, Lax Epimorphisms and Effective Descent for Split Fibrations

Abstract

For any suitable base category V, we find that V-fully faithful lax epimorphisms in V-Cat are precisely those V-functors F:A→B whose induced V-functors CauchyF:CauchyA→CauchyB between the Cauchy completions are equivalences. For the case V=Set, this is equivalent to requiring that the induced functor CAT(F,Cat) between the categories of split (op)fibrations 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 these observations 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.