Cauchy Completeness, Lax Epimorphisms and Effective Descent for Split Fibrations

TL;DR

This paper characterizes fully faithful lax epimorphisms in enriched categories via Cauchy completions and relates them to effective descent morphisms, providing new insights into fibrations and descent theory in category theory.

Contribution

It establishes a new characterization of fully faithful lax epimorphisms using Cauchy completions and links these to effective descent morphisms in the context of split fibrations.

Findings

Fully faithful lax epimorphisms correspond to equivalences on Cauchy completions.

Effective descent morphisms are characterized via codescent factorization.

Connections between fibrations, Cauchy completions, and descent theory are elucidated.

Abstract

For any suitable base category $\mathcal{V} $, we find that $\mathcal{V} $-fully faithful lax epimorphisms in $\mathcal{V} $-$\mathsf{Cat} $ are precisely those $\mathcal{V}$-functors $F \colon \mathcal{A} \to \mathcal{B}$ whose induced $\mathcal{V} $-functors $\mathsf{Cauchy} F \colon \mathsf{Cauchy} \mathcal{A} \to \mathsf{Cauchy} \mathcal{B} $ between the Cauchy completions are equivalences. For the case $\mathcal{V} = \mathsf{Set} $, this is equivalent to requiring that the induced functor $\mathsf{CAT} \left( F,\mathsf{Cat}\right) $ 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 $\mathcal{F}$ to the study of the codescent factorization, we find that these observations on fully faithful lax epimorphisms provide us with a characterization of (effective) $\mathcal{F}$-descent morphisms in the category of small categories $\mathcal{Cat}$; 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.