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

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.