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

Nick Hu, Jamie Vicary

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.

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 , 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.

Paper Structure

This paper contains 18 sections, 16 theorems, 27 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related work
  3. Outline
  4. Acknowledgements
  5. String diagrams and the bicategory of profunctors
  6. Introduction
  7. Planar monoidal categories
  8. Braiding and symmetry
  9. Autonomous categories
  10. Internal string diagrams
  11. The traced pseudomonoid presentation
  12. Traced monoidal categories
  13. Braided autonomous categories are traced
  14. On traced *-autonomous categories
  15. *-Autonomous categories
  16. ...and 3 more sections

Key Result

Lemma 1

[lemma]prof-embedding For each functor $F\colon \mathcal{C} \to \mathcal{D}$, there are associated profunctors $F_*\colon {\mathcal{C}}^\mathrm{op} \times \mathcal{D} \to \mathop{\mathrm{\mathbf{Set}}}\nolimits$ and $F^*\colon {\mathcal{D}}^\mathrm{op} \times \mathcal{C} \to \mathop{\mathrm{\mathbf{ Either mapping extends to an injective fully faithful pseudofunctor borceuxHandbookCategoricalAlgeb

Figures (1)

  • Figure 1: Examples of the internal string diagram formalism.

Theorems & Definitions (48)

  • Definition 1: Bicategory of Profunctors borceuxHandbookCategoricalAlgebra1994
  • Lemma 1
  • Theorem 1: borceuxHandbookCategoricalAlgebra1994
  • Definition 2
  • Definition 3
  • Lemma 2
  • Definition 4: Free right-adjoint extension
  • Example 1
  • Lemma 3
  • Lemma 4
  • ...and 38 more