Traced Monoidal Categories as Algebraic Structures in Prof

TL;DR

This paper introduces a new algebraic framework for understanding traced monoidal categories using monoidal bicategories, enabling advanced reasoning and revealing new equivalences and conditions in traced autonomous categories.

Contribution

It defines traced pseudomonoids in monoidal bicategories, providing a novel algebraic characterization of Cauchy complete traced monoidal categories and applying graphical calculus for deeper insights.

Findings

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

Described conditions under which traced *-autonomous categories become autonomous.

Demonstrated the utility of graphical calculus in reasoning about traced monoidal structures.

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 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 ⊗-trace and the right ⅋-trace, and describing a new condition under which traced ∗-autonomous categories become autonomous.