Traced monoidal categories as algebraic structures in $\mathbf{Prof}$

TL;DR

This paper introduces a new algebraic framework for traced monoidal categories using pseudomonoids in the bicategory of profunctors, enabling advanced reasoning and new results in the theory of traced categories.

Contribution

It characterizes Cauchy complete traced monoidal categories as algebraic structures in $ extbf{Prof}$, providing a novel approach to their analysis and applications.

Findings

Graphical calculus for monoidal bicategories is used to reason about traces.

Proves a new equivalence between left and right traces in traced $*$-autonomous categories.

Identifies conditions under which traced $*$-autonomous categories become autonomous.

Abstract

We define a traced pseudomonoid as a pseudomonoid in a monoidal bicategory equipped with extra structure, giving a new characterisation of Cauchy complete traced monoidal categories as algebraic structures in $\mathbf{Prof}$, the monoidal bicategory of profunctors. This enables reasoning about the trace using the graphical calculus for monoidal bicategories, which we illustrate in detail. We apply our techniques to study traced $*$-autonomous categories, proving a new equivalence result between the left $\otimes$-trace and the right $\unicode{8523}$-trace, and describing a new condition under which traced $*$-autonomous categories become autonomous.