On effective descent $\mathcal V$-functors and familial descent morphisms

Abstract

We study effective descent $ \mathcal V $-functors for cartesian monoidal categories $ \mathcal V $ with finite limits. This study is carried out via the properties enjoyed by the $2$-functor $ \mathcal V \mapsto \mathsf{Fam}(\mathcal 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 $ \mathsf{Fam}(\mathcal V) $, we briefly study those epimorphisms when $ \mathcal V $ is a regular category.