Date Available

5-10-2023

Year of Publication

2023

Degree Name

Doctor of Philosophy (PhD)

Document Type

Doctoral Dissertation

College

Arts and Sciences

Department/School/Program

Mathematics

First Advisor

Dr. Kate Ponto

Abstract

Familiar constructions like the trace of a matrix and the Euler characteristic of a closed smooth manifold are generalized by a notion of trace of an endomorphism of a dualizable object in a bicategory equipped with a piece of additional structure called a shadow functor. Another example of this bicategorical trace, in the form of maps between Hochschild homology of bimodules, appears in a 1987 paper by Joseph Lipman, alongside a more mysterious ”cotrace” map involving Hochschild cohomology. Putting this cotrace on the same category-theoretic footing as the trace has led us to propose a ”bicategorical cotrace” in a closed bicategory with a ”coshadow functor.” The program of bicategorical shadows and traces aims to unify seemingly disparate pieces of mathematics underneath a common conceptual framework; by adding notions of coshadow and cotrace to this machinery, we have drawn Lipman’s residues and (co)traces into this framework and made progress toward describing 2-representations and 2-characters in a way that parallels the application of traces to ordinary group representations. We begin by reviewing the theory of duality and trace in symmetric monoidal categories and in bicategories, and we discuss the features of closed bicategories that will be needed to develop a theory of cotraces. We then motivate and define bicategorical coshadows and cotraces and proceed to establish several important properties of these constructions. We also prove a very general interplay between traces and cotraces in a closed bicategory with compatible shadows and coshadows. Finally, we discuss applications of the bicategorical cotrace to Lipman’s residues and traces and to Ganter and Kapranov’s study of 2-representations and 2-characters.

Digital Object Identifier (DOI)

https://doi.org/10.13023/etd.2023.205

Funding Information

This work was supported by a summer research fellowship from the University of Kentucky mathematics department in 2021.

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