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nLab > Latest Changes: commutative monoidal category

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  1. Page created, but author did not leave any comments.

    Jade Master

    v1, current

  2. Someone told me that line bundles of affine schemes (I think) are related to commutative monoidal categories but I’m not sure how. If someone knows please add it :)

    Jade Master

    v1, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 1st 2019

    I have touched the first paragraphs, adding hyperlinks (for instance to permutative monoidal category) and making some slight changes. Please check if you can live with it.

    I also looked for, found and added a reference. It turns out to be by you! :-)

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorjademaster
    • CommentTimeDec 1st 2019
    Looks good! Thanks Urs.
    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 1st 2019

    By the way, since the nLab doesn’t go by a Wikipedia-style “neutral point of view”-paradigm, but instead by the idea that if you are the expert on a subject, then we want you to not shy away from sharing your insights, you are invited to add references to your own articles. In fact you should not put that burden on other shoulders! Like mine in this case.

    That’s just the evident consequence of you editing an entry on a given subject in the first place. If you have more to say on that subject, and if you did say it in a preprint or publication, then let us know where to find this. In converse, if your publications are inappropriate for linking on the nLab, then your edits to the nLab must by extension be inappropriate, too, and then we you will hear from the steering committee anyway. :-)

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeDec 1st 2019

    Re #2, possibly the reference is Doctrines of Algebraic Geometry? (Is there a way to link directly to a “published” web from here?)

    • CommentRowNumber7.
    • CommentAuthorJohn Baez
    • CommentTimeDec 1st 2019

    Added alternative definition of commutative monoidal category.

    diff, v3, current

    • CommentRowNumber8.
    • CommentAuthorJohn Baez
    • CommentTimeDec 1st 2019

    Added examples, and a conjecture about when symmetric monoidal category are equivalent to commutative ones.

    diff, v3, current

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 1st 2019

    John, I haven’t thought about this question, but where is it written down the claim about the category of line bundles? I would think that might give inspiration.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeDec 2nd 2019

    Moved the conjecture to a separate section, and remarked that one direction of it is certainly true.

    diff, v4, current

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeDec 2nd 2019

    Regarding the conjecture, suppose we consider the groupal case. Section 2 of Johnson-Orsono shows that a symmetric groupal groupoid (i.e. Picard groupoid) is equivalent to a skeletal and strictly-associative one, and that the symmetry is determined by a cocycle c:G×G→Mc : G\times G \to M (where GG is the group of objects and MM the group of automorphisms of the unit) such that c(x,y+z)=c(x,y)+c(x,z)c(x,y+z) = c(x,y) + c(x,z) and c(x,y)=−c(y,x)c(x,y) = - c(y,x). Two such cocycles c,c′c,c' determine equivalent Picard groupoids if they are “cohomologous” in the sense that there is a k:G×G→Mk:G\times G\to M such that k(x,0)=k(0,y)=0k(x,0)=k(0,y)=0 and k(x+y,z)+k(x,y)=k(x,y+z)+k(y,z)k(x+y,z) + k(x,y) = k(x,y+z)+k(y,z) and c(x,y)−c′(x,y)=k(x,y)−k(y,x)c(x,y) - c'(x,y) = k(x,y)-k(y,x). Now a commutative groupal groupoid would have c(x,y)=0c(x,y)=0 for all x,yx,y, while the condition given in the conjecture says c(x,x)=0c(x,x)=0 for all xx. So the question in that case is, if c(x,x)=0c(x,x)=0 for all xx, is cc necessarily cohomologous to 00? To say that cc is cohomologous to 00 means there is a kk satisfying the cocycle conditions and with c(x,y)=k(x,y)−k(y,x)c(x,y) = k(x,y)-k(y,x). This last condition certainly implies that c(x,x)=0c(x,x)=0, but we knew that already. It’s not at all clear to me how assuming that c(x,x)=0c(x,x)=0 helps in constructing such a kk, but I haven’t thought about it for more than a minute.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeDec 2nd 2019
    • (edited Dec 2nd 2019)

    I changed the syntax of the hyperlink to James Dolan’s page to the more robust double square bracket syntax.

    I also changed “we conjecture” to “one might conjecture”.

    Maybe you want to have it say: “In Baez-Master 18 it is conjectured…”, which would work. But first person pronouns don’t really work on public wiki pages.

    diff, v5, current

    • CommentRowNumber13.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 2nd 2019

    But first person pronouns don’t really work on public wiki pages.

    I tend to agree, and yet the nLab has quite a few of them (many from Toby’s hand).

    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeDec 2nd 2019

    Sometimes I write something like “I (Mike Shulman) conjecture that…”. I think it’s reasonable to write down our conjectures in our lab book, as long as we’re clear about who we are.

    • CommentRowNumber15.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 2nd 2019

    I do that too, and also think that’s okay. But I’ve noticed that Toby doesn’t identify himself in many cases.

    • CommentRowNumber16.
    • CommentAuthorJoe Moeller
    • CommentTimeDec 2nd 2019

    added redirects

    diff, v6, current

    • CommentRowNumber17.
    • CommentAuthorJ-B Vienney
    • CommentTimeMar 17th 2023

    The conjecture “A symmetric monoidal category is symmetric monoidally equivalent to a commutative monoidal category iff all its self-braidings are identity morphisms.” is proved in the added reference.

    diff, v7, current

    • CommentRowNumber18.
    • CommentAuthorvarkor
    • CommentTimeMar 17th 2023

    Note that the proof of the conjecture relies upon a total ordering of objects.

    diff, v8, current

  19. Add reference to Meseguer and Montanari

    Mario Román

    diff, v9, current

    • CommentRowNumber20.
    • CommentAuthorvarkor
    • CommentTimeSep 25th 2024

    Added redirects for “strict symmetric monoidal category”.

    diff, v12, current

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