The familial nature of enrichment over virtual double categories

TL;DR

This paper explores how enriching over virtual double categories broadens the scope of examples and enhances formal properties, including a new construction of the 2-category of enriched categories.

Contribution

It demonstrates that working over virtual double categories yields better formal properties and introduces a families construction for these categories.

Findings

The 2-functor to categories enriched over virtual double categories is a parametric right 2-adjoint.

Enrichment over virtual double categories is more inclusive and formally robust.

A formal families construction for virtual double categories is developed.

Abstract

Originally enriched categories were defined over a monoidal category, but it was gradually realized that important examples can only be included when one enriches over more general structures such as bicategories and virtual double categories. We show that, as well as allowing more examples, working over virtual double categories also gives better formal properties. We study the 2-functor sending a virtual double category to the 2-category of categories enriched over it. We show that this is a parametric right 2-adjoint, and in fact is familial. We also show how a ``families construction'' for virtual double categories can be used to give a formal construction of the 2-category of categories enriched over a virtual double category.