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

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

    diff, v6, current

    • 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.

    diff, v9, current

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