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nLab > Latest Changes: compact closed category

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  1. I added a note to compact closed category on the fact that the inclusion from compact closed categories into SMCCs has a left adjoint, pointing to an article by Day where he describes the free compact closed category over a closed symmetric monoidal category as a localization. Question: this left adjoint is not full, but I believe it is faithful – does anyone know how to prove that?

  2. actually, I’m not so sure the functor is faithful, because I don’t see what rules out Kelly & Mac Lane’s counterexample from “Coherence in closed categories”: the morphism [[[A,I],I],I]→[[[A,I],I],I] defined as the composition [[[A,I],I],I]→[ηA,id][A,I]→η[A,I][[[A,I],I],I] is not the identity in an arbitrary SMCC C, but if I’m not mistaken it must be mapped to the identity in the free compact closed category over C. Perhaps my question was not so relevant.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 13th 2015

    Actually, I wasn’t even sure what you meant. Reading your last comment, I think you may have meant that the unit of the adjunction M→UFM, where M is an smcc, F is the free (2-)functor, and U is the forgetful (2-)functor from compact closed cats to smcc’s, is faithful, and that’s what you’re calling into question now with the Kelly-Mac Lane example. (The comment in #1 made me wonder whether you meant F itself was faithful in some 2-categorical sense.)

  4. Oh, you’re right, that’s what I actually had in mind. And in thinking that the unit M→UFM was faithful, I was overlooking the Kelly-Mac Lane example.

  5. add precisions about FinVect being compact closed: this depends on the monoidal structure chosen!

    Antonin Delpeuch

    diff, v30, current

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 4th 2019

    Added doi links for the references.

    diff, v32, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 15th 2023

    added pointer to:

    diff, v41, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeSep 15th 2023

    and this one:

    diff, v41, current

  9. fixed a typo

    Adam Nemecek

    diff, v43, current

    • CommentRowNumber10.
    • CommentAuthorvarkor
    • CommentTimeFeb 22nd 2024

    Mention that compact closed categories are self-dual.

    diff, v44, current

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 23rd 2024

    Added proof that a compact closed category with binary products that distribute over binary coproducts is thin, as discussed on CT Zulip.

    diff, v45, current

    • CommentRowNumber12.
    • CommentAuthorvarkor
    • CommentTimeApr 20th 2024

    Add a cross-reference to pregroup grammar.

    diff, v47, current

    • CommentRowNumber13.
    • CommentAuthorJohn Baez
    • CommentTimeAug 15th 2024

    Clarified description of how compact closed categories are a special case of closed categories.

    diff, v48, current

    • CommentRowNumber14.
    • CommentAuthorJohn Baez
    • CommentTimeAug 15th 2024

    In section “Relation to symmetric monoidal closed categories”, I added a remark about a false characterization of compact closed categories among symmetric monoidal closed ones, which various experts have tripped over.

    diff, v48, current

    • CommentRowNumber15.
    • CommentAuthorJohn Baez
    • CommentTimeAug 15th 2024

    Added more detail on how delooping of a commutative monoid is a compact closed category.

    diff, v49, current

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