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nLab > Latest Changes: promonoidal category

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  1.  
    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeMay 15th 2012

    I moved the definition of promonoidal categories from Day convolution to promonoidal category, and expanded on it a bit.

    • CommentRowNumber2.
    • CommentAuthorNoam_Zeilberger
    • CommentTimeApr 2nd 2015
    • (edited Apr 2nd 2015)

    I noticed that the definition of promonoidal category given here is op’d from the one in Day’s original paper (and the one with Panchadcharam and Street), in the sense that a Day-promonoidal structure on CC corresponds to a nlab-promonoidal structure on C opC^{op}. I imagine this reversal has to do with the convention for the direction of profunctors. Should we care? I don’t think it matters for the promonoidal structure generated by a monoidal category (since a monoidal structure on CC transports to a monoidal structure on C opC^op), but on the other hand, another example Street gives is of the promonoidal structure generated by a biclosed category, with P(A,B,C)=Hom(B,A∖C)=Hom(A,C/B)P(A,B,C) = Hom(B, A\setminus C) = Hom(A, C/B). That wouldn’t work with the current definition in the article, since the existence of a biclosed structure is not preserved under op’ing.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 2nd 2015

    Good catch, Noam; that’s a little annoying. I think we should care, and I’d say we should change over to the standard Day definition, or at the very least leave a remark.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeApr 3rd 2015

    What about the equivalence between promonoidal structures on CC and biclosed monoidal structures on V C opV^{C^{op}}?

    • CommentRowNumber5.
    • CommentAuthorNoam_Zeilberger
    • CommentTimeApr 3rd 2015
    • (edited Apr 3rd 2015)

    Looking at (Day 1974) as well as the (Day & LaPlaza 1978) paper cited at closed category, it seems the idea is that a (Day-)promonoidal structure on the closed category CC is used to construct a biclosed monoidal structure on V CV^C, and then CC is embedded into contravariant presheaves over a small subcategory A⊂V CA \subset V^C. So perhaps it is just worth adding a note that a closed structure on CC induces a (nlab-)promonoidal structure on C opC^op, and that this is op’d from Day’s original convention?

  6. (and that actually clarifies for me why you have to “take presheaves twice” in the construction of a biclosed monoidal category over a closed category – when one suffices for getting a biclosed monoidal category over a monoidal category – so I see the advantage of the current convention used in the nlab article.)

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeApr 3rd 2015

    More generally, any multicategory structure on CC induces an nlab-promonoidal structure on C opC^{op}.

    I find it very pleasing when using the convention where ProfProf is equivalent to the 2-category of presheaf categories and cocontinuous functors, which makes it utterly obvious that a promonoidal structure on CC is the same as a nice monoidal structure on V C opV^{C^{op}}.

  8. So I went ahead and added a little note about the relationship to Day’s definition. (Also note that he had actually first called these “premonoidal categories”, but sometime between 1970 and 1974 he switched to “promonoidal” and reformulated the definition in terms of profunctor composition.)

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeApr 6th 2015

    Thanks!

  10. Added the diagrams corresponding to the triangle and pentagon identities for promonoidal categories. (First time making a non-trivial contribution to the nLab; I hope everything is okay.)

    diff, v10, current

    • CommentRowNumber11.
    • CommentAuthorDELETED_USER_2018
    • CommentTimeMay 10th 2020
    • (edited Apr 11th 2023)
    [deleted]
    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 10th 2020

    I have fixed the numbered lists with the diagrams. The trick is to have all content of a numbered item be indented by exactly 3 whitespace.

    (Single lines of running text wrapping the edit pane width just look like they are not fully indented, but as long as they don’t contain an explicit “carriage return” they are.)

    diff, v11, current

    • CommentRowNumber13.
    • CommentAuthorGuest
    • CommentTimeMay 10th 2020

    For a more compact way of presenting the pentagon and triangle identities for a promonoidal category, see diagrams (11.47) and (11.51), and the paragraph preceding them, in Ross Street’s Skew-closed categories.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeMay 10th 2020
    • (edited May 10th 2020)

    I have added hyperlinks to a bunch of further keywords.

    (Noticed that we really ought to make triangle identity a disambiguation page…)

    diff, v11, current

    • CommentRowNumber15.
    • CommentAuthorDELETED_USER_2018
    • CommentTimeMay 10th 2020
    • (edited Apr 11th 2023)

    [deleted]

    • CommentRowNumber16.
    • CommentAuthormaxsnew
    • CommentTimeApr 6th 2023

    The idea section says that a promonoidal category is a categorification of a boolean algebra. Does anyone know what this is supposed to mean? This is a categorification of many things, and doesn’t seem particularly similar to boolean algebras, or any more similar to boolean algebras than just semi-lattices.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeApr 6th 2023

    Checking the page history, this statement as added by “Anonymous” in revision 3, back in 2013.

    I suppose we should just delete it.

    • CommentRowNumber18.
    • CommentAuthormaxsnew
    • CommentTimeApr 7th 2023

    Remove boolean algebra comment from idea section

    diff, v14, current

    • CommentRowNumber19.
    • CommentAuthorvarkor
    • CommentTimeDec 22nd 2023

    Added a reference to Lax monoids, pseudo-operads, and convolution.

    diff, v16, current

    • CommentRowNumber20.
    • CommentAuthorvarkor
    • CommentTimeMar 4th 2024

    Clarify the characterisation of promonoidal categories in terms of multicategories.

    diff, v17, current

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