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nLab > Latest Changes: rigid monoidal category

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  1.  
    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 20th 2021

    Added:

    Free rigid monoidal categories

    The inclusion of the 2-category of monoidal categories into the 2-category of rigid monoidal categories admits a left 2-adjoint functor LL.

    Furthermore, the unit of the adjunction is a strong monoidal fully faithful functor, i.e., any monoidal category CC admits a fully faithful strong monoidal functor C→L(C)C\to L(C), where L(C)L(C) is a rigid monoidal category.

    See Theorems 1 and 2 in Delpeuch \cite{Delpeuch}.

    diff, v17, current

  2. Started making the page more readable - not at all done

    Milo Moses

    diff, v19, current

  3. Added the example of vector spaces

    Milo Moses

    diff, v19, current

  4. Added “adjointness=dual” details

    Milo Moses

    diff, v19, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2023

    Thanks!

    Looking at the entry now, I have reorganized a little: Moved the previous section “Remarks” and your new section on string diagrams from after the Examples-section to subsections inside the Definition-section, where they seem to better belong.

    While I was at it (and unrelated to your edit), I looked through the old Examples-section (here) and made some adjustments on wording and hyperlinking.

    I think the examples on the category of finite-dimensional vector spaces (here) was missing the finite-dimensionality clause in a couple of crucial places.

    and I noitce that the Section “Endofunctor categories” is somewhat misleading in its title and never quite gets around to naming the actual rigid monoidal subcategory of the full endofunctor category that is an example. I have made some adjustments, but this still deserves attention.

    diff, v21, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2023

    In the old section “Free rigid monoidal catgeories” (here) I have

    fixed the sentence about which 2-category is embedded in which one,

    fixed the intended hyperlink to adjoint 2-functor

    added more hyperlinks overall

    and added the publication data to the reference

    diff, v21, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2023

    I fact, looking closer at the paragraph on the example of “endofunctor categories”, I found it awkward in how it spoke about adjunction (co)units in a rounabout way without naming them and using ill-typed notation for them.

    So I have taken the liberty of deleting this and replacing it simply by a paragraph saying straight away that adjoint endofunctors are the dualizable objects with (co)evaluation being their (co)unit.

    Even so, the example still does not fit well here at rigid monoidal category. This is really an example of dualizable objects…. and now that I write this I decide to delete it here entirely and move it to there.

    If anyone feels strongly otherwise, please say and we can try to find out what the intent of the example here was.

    diff, v22, current

  8. Added punctuation.

    Benjamin Andersson

    diff, v25, current

1 to 8 of 8

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