Bicategorical Traces and Cotraces

TL;DR

This paper extends the concept of trace from matrices to dualizable objects in symmetric monoidal categories, introducing bicategorical cotraces within bicategories with shadows, linking to Hochschild (co)homology and 2-representations.

Contribution

It proposes a new notion of bicategorical cotrace in bicategories with coshadows, generalizing trace concepts to broader categorical contexts.

Findings

Introduces bicategorical cotrace framework for dualizable objects.

Connects bicategorical traces to Hochschild (co)homology.

Relates bicategorical structures to 2-representations and 2-characters.

Abstract

The familiar trace of a square matrix generalizes to a trace of an endomorphism of a dualizable object in a symmetric monoidal category. To extend these ideas to other settings, such as modules over non-commutative rings, the trace can be generalized to a bicategory equipped with additional structure called a shadow. We propose a notion of bicategorical cotrace of certain maps involving dualizable objects in a closed bicategory equipped with a coshadow, and we use this framework to draw connections to work of Lipman on residues and traces with Hochschild (co)homology, and to work of Ganter and Kapranov on 2-representations and 2-characters.