Free globularily generated double categories

TL;DR

This paper introduces a construction for free globularily generated double categories that canonically extends bicategories with vertical morphisms, enabling new formalizations in operator algebra theory.

Contribution

It presents the first construction of free globularily generated double categories and applies it to extend key concepts in operator algebras.

Findings

Constructed the free globularily generated double category from bicategories.

Applied the construction to extend Haagerup's standard form and Connes fusion.

Provided insights into length, free products, and internalization problems.

Abstract

This is the first part of a two paper series studying free globularily generated double categories. In this first installment we introduce the free globularily generated double category construction. The free globularily generated double category construction canonically associates to every bicategory together with a possible category of vertical morphisms, a double category fixing this set of initial data in a free and minimal way. We use the free globularily generated double category to study length, free products, and problems of internalization. We use the free globularily generated double category construction to provide formal functorial extensions of the Haagerup standard form construction and the Connes fusion operation to inclusions of factors of not-necessarily finite Jones index.