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

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

]]>Thanks!

]]>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.

]]>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*.

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.

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