A new construction relating enriched categories and internal ones in an extensive ambient

TL;DR

This paper introduces a novel construction linking enriched categories with internal categories within extensive ambient categories, utilizing a functorial process that generalizes idempotent splitting.

Contribution

It presents a new functorial construction that relates enriched categories to internal categories in extensive ambient categories, extending existing methods with a focus on idempotent splitting.

Findings

The construction is functorial and left adjoint to a known functor.

It applies to a specific class of internal categories with canonical idempotent splitting.

The method involves a size restriction to ensure well-definedness.

Abstract

A new construction to associate an internal category to an enriched one is presented. The key concept is that of extensive ambient category, and the construction follows the one that associates a category whose idempotents split to a given one. The association turns out to be functorial and left adjoint to an already known one when we restrict to a particular class of internal categories whose idempotents split in some canonical way and impose a size restriction.