Reflection on the enrichment of double categories

TL;DR

This paper explores the concept of enriching double categories, aiming to clarify and extend the understanding of how double categories can be enriched, especially in relation to internal categories and their generalizations.

Contribution

It introduces a new perspective on enriching double categories, addressing limitations of existing internal category approaches and broadening the conceptual framework.

Findings

Enrichment by double categories helps incorporate internal categories of enriched categories.

Existing internal category approaches are insufficient for thin or framed bicategories.

The new approach relaxes previous constraints, enabling a more comprehensive understanding of double categories.

Abstract

Enrichment and internal categories are two different way to generalize the notion of category. As such, enriching double categories (which are categories internal to Cat) is not a clear concepts. One can look at the internal categories of the category of C-enriched categories or the internal C-category in Cat. This concepts, unfortunately, are not powerful enough, even thin (posetal) double categories can't be seen this way. However, an orthogonal investigation, that of the enrichment by a double category, will help relaxing the internal C-category to a incorporate the internal categories of the category of C-enriched categories, as well as thin double categories or framed bicategories.