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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 18th 2014

created traced monoidal category with a bare minimum

I would have sworn that we already had an entry on that, but it seems we didn’t. If I somehow missed it , let me know and we need to fix things then.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJan 19th 2014

added a brief paragraph on the relation to compact closed categories:

Given a traced monoidal category $\mathcal{C}$, there is a free construction completion of it to a compact closed category $Int(\mathcal{C})$ (Joyal-Street-Verity 96):

the objects of $Int(\mathcal{C})$ are pairs $(A^+, A^-)$ of objects of $\mathcal{C}$, a morphism $(A^+ , A^-) \to (B^+ , B^-)$ in $Int(\mathcal{C})$ is given by a morphism of the form $A^+\otimes B^- \longrightarrow A^- \otimes B^+$ in $\mathcal{C}$, and composition of two such morphisms $(A^+ , A^-) \to (B^+ , B^-)$ and $(B^+ , B^-) \to (C^+ , C^-)$ is given by tracing out $B^+$ and $B^-$ in the evident way.

Copied the same paragraph over to compact closed category.

1. I added a reference to the characterization of traces in cartesian monoidal categories (by Hasegawa and Hyland), and then because this page was still missing a definition I pulled in the definition (for symmetric monoidal categories) from Hasegawa.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeNov 19th 2014

Thanks!

2. copied in and adapted definition of the canonical trace on a compact closed category.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeNov 28th 2019

Added Plotkin’s observation that not every monoidal category can even be mapped faithfully into a traced monoidal category.