Bicategorical traces and cotraces

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.The author is grateful to Jonathan Campbell for bringing to our attention the paper [Lip87] that motivated this work, to Cary Malkiewich for suggesting a better system of string diagrams and thereby helping to formulate cyclicity for cotraces, to Millie Rose Deaton, Niles Johnson, Inbar Klang, and Jordan Sawdy for helpful conversations, to