Traced monoidal categories as algebraic structures in $\mathbf{Prof}$
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866911791853862912 |
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| author | Hu, Nick Vicary, Jamie |
| author_facet | Hu, Nick Vicary, Jamie |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_00589 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Traced monoidal categories as algebraic structures in $\mathbf{Prof}$ Hu, Nick Vicary, Jamie Category Theory 18M10, 18M30, 18D60, 18D70 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. |
| title | Traced monoidal categories as algebraic structures in $\mathbf{Prof}$ |
| topic | Category Theory 18M10, 18M30, 18D60, 18D70 |
| url | https://arxiv.org/abs/2109.00589 |