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 $\ast$-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 $\ast$-autonomous categories become autonomous.

References in corpus (6)

• A survey of graphical languages for monoidal categories
• The Classification of Two-Dimensional Extended Topological Field Theories
• Compact Closed Bicategories
• Modular categories as representations of the 3-dimensional bordism 2-category
• Quasistrict symmetric monoidal 2-categories via wire diagrams
• Coherence for Frobenius pseudomonoids and the geometry of linear proofs

Cited by in corpus (2)

• Coend Optics for Quantum Combs
• Collages of String Diagrams