This is the first part of a two paper series studying free globularly generated double categories. In this first installment we introduce the free globularly generated double category construction. The free globularly 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 globularly generated double category to study length, free products, and problems of internalization. We use the free globularly 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.
Keywords: Category, double category, bicategory, von Neumann algebra, representation, fusion, tensor category, length
2010 MSC: 18B30,18D05,18F15
Theory and Applications of Categories, Vol. 34, 2019, No. 42, pp 1343-1385.