Enriched categories as a free cocompletion

TL;DR

This paper develops a theory of bicategories enriched in monoidal bicategories, generalizing classical enriched category theory, and characterizes the free cocompletion process with a universal property involving enriched categories and modules.

Contribution

It extends enriched category theory to bicategories enriched in monoidal bicategories and describes their free cocompletion via weighted bicolimits, with a universal property for enriched equipments.

Findings

Established a theory of bicategories enriched in monoidal bicategories.

Characterized the free cocompletion of enriched bicategories using weighted bicolimits.

Provided a universal property for the construction involving enriched categories and modules.

Abstract

This paper has two objectives. The first is to develop the theory of bicategories enriched in a monoidal bicategory -- categorifying the classical theory of categories enriched in a monoidal category -- up to a description of the free cocompletion of an enriched bicategory under a class of weighted bicolimits. The second objective is to describe a universal property of the process assigning to a monoidal category V the equipment of V-enriched categories, functors, transformations, and modules; we do so by considering, more generally, the assignation sending an equipment C to the equipment of C-enriched categories, functors, transformations, and modules, and exhibiting this as the free cocompletion of a certain kind of enriched bicategory under a certain class of weighted bicolimits.