Internalizing decorated bicategories: The globularily generated condition

TL;DR

This paper introduces a formal framework for the existence of double categories with given horizontal bicategories and object categories, focusing on the globularily generated condition as a key minimal criterion.

Contribution

It establishes the equivalence between the existence of internalizations and globularily generated solutions, providing a foundational approach to internalizing decorated bicategories.

Findings

Existence of internalizations is equivalent to the existence of globularily generated solutions.

The globularily generated condition is the minimal criterion for solutions.

The paper formalizes the problem of internalizing decorated bicategories.

Abstract

This is the first part of a series of papers studying the problem of existence of double categories for which horizontal bicategory and object category are given. We refer to this problem as the problem of existence of internalizations for decorated bicategories. We establish a formal framework within which the problem of existence of internalizations can be correctly formulated. Further, we introduce the condition of a double category being globularily generated. We prove that the problem of existence of internalizations for a decorated bicategory admits a solution if and only if it admits a globularily generated solution, and we prove that the condition of a double category being globularily generated is precisely the condition of a solution to the problem of existence of internalizations for a decorated bicategory being minimal. The study of the condition of a double category being globularily generated will thus be pivotal in our study of the problem of existence of internalizations.