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nLab > Latest Changes: traced monoidal category

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  1.  
    • 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.

  3. 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!

  5. 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

  7. Provided an example of a monoidal category which does not satisfy cancellation.

    Robin Kaarsgaard

    diff, v10, current

  8. Can’t get the latex to work, reverting change for now.

    Robin Kaarsgaard

    diff, v10, current

    • CommentRowNumber9.
    • CommentAuthorSam Staton
    • CommentTimeMay 16th 2022

    Added trace in fdvect

    diff, v11, current

    • CommentRowNumber10.
    • CommentAuthorSam Staton
    • CommentTimeMay 17th 2022

    Example of trace of profunctors

    diff, v12, current

    • CommentRowNumber11.
    • CommentAuthorSam Staton
    • CommentTimeMay 17th 2022

    trace in partial functions

    diff, v12, current

    • CommentRowNumber12.
    • CommentAuthorSam Staton
    • CommentTimeMay 17th 2022

    Trace in category of pointed cpo’s

    diff, v12, current

    • CommentRowNumber13.
    • CommentAuthormaxsnew
    • CommentTimeAug 19th 2022

    Add the relation to programming to the idea section, and link to a page on fixed point operators (to be made)

    diff, v15, current

  14. Adding the related notion of “feedback category” (or “category-with-feedback”, or “category with delayed-feedback”) that appears in the work of Katis, Sabadini and Walters.

    Mario Román

    diff, v16, current

    • CommentRowNumber15.
    • CommentAuthorJ-B Vienney
    • CommentTimeMay 28th 2023
    • (edited May 28th 2023)

    An axiom (sliding) was unclear due to a lack of parentheses.

    diff, v18, current

    • CommentRowNumber16.
    • CommentAuthorJ-B Vienney
    • CommentTimeJun 23rd 2024
    • (edited Jun 23rd 2024)

    Added the example of the preordered set of divisibility of a cancellative commutative monoid.

    diff, v20, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeNov 5th 2024
    • (edited Nov 5th 2024)

    In reaction to a comment here I have swapped the symbols “hh” and “gg” in the line where they are declared (first line of the “Thightening” axiom here).

    But I haven’t checked anything myself – authors of this page should please have a look.

    diff, v23, current

1 to 17 of 17

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