Scalar enrichment and cotraces in bicategories

Abstract

It is known that every monoidal bicategory has an associated braided monoidal category of scalars. In this thesis we show that every monoidal bicategory, which is closed both monoidally and compositionally, can be enriched over the monoidal 2-category of scalar-enriched categories. This enrichment provides a number of key insights into the relationship between linear algebra and category theory. The enrichment replaces every set of 2-cells with a scalar, and we show that this replacement can be given in terms of the cotrace, first defined by Day and Street in the context of profunctors. This is analogous to the construction of the Frobenius inner product between linear maps, which is constructed in terms of the trace of linear maps. In linear algebra it is also possible to define the trace in terms of the Frobenius inner product. We show that the cotrace can be defined in terms of the enrichment, and in doing so we prove that the cotrace is an enriched version of the `categorical trace' studied by Ganter and Kapranov, and Bartlett. Thus, we unify the concept of a categorical trace with the concept of a cotrace. Finally, we study the relationship between the trace and the cotrace for compact closed bicategories. We show that the trace and cotrace have a structured relationship and share many of the properties of the linear trace including -- but not limited to -- dual invariance and linearity. Motivating examples are given throughout. We also introduce a decorated string diagram language to simplify some of the proofs.