On effective descent [formula omitted]-functors and familial descent morphisms

Abstract

We study effective descent V-functors for cartesian monoidal categories V with finite limits. This study is carried out via the properties enjoyed by the 2-functor V↦Fam(V), results about effective descent of bilimits of categories, and the fact that the enrichment 2-functor preserves certain bilimits. Since these results rely on an understanding of (effective) descent morphisms in Fam(V), we carefully study these morphisms in free coproduct completions. Finally, we provide refined conditions when V is a regular category.