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nLab > Latest Changes: operad

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  1.  
    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 23rd 2010

    Wrote a section on the associated monad at operad, in terms of the framework introduced under the section titled Preparation.

    • CommentRowNumber2.
    • CommentAuthorHarry Gindi
    • CommentTimeMay 23rd 2010

    @Todd: Perhaps we should move the detailed conceptual treatment to a separate page (including this part that you wrote just now). I was thinking of writing out some of the easy proofs of the statements, and it seems like that section is growing independently of the rest of the article. It seems like a good candidate for splitting, but I defer to your judgement.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 23rd 2010

    Anyone else have an opinion about Harry’s suggestion? As for myself, I’ll think about it and maybe sleep on it.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeMay 23rd 2010

    I’m not sure why that section is called “detailed conceptual treatment” – it seems to me to be just one way of giving the definition. The previous definition is just as adequate, modulo the missing coherence diagrams. Another way of giving the definition is in terms of generalized multicategories, which is not discussed much at operad. (By the way, operad and multicategory and even globular operad were surprisingly lacking in links to generalized multicategory, so I added some.) I could see the argument for splitting it off, especially if you’re going to add detailed proofs; in line with the “zoomability” philosophy the article “operad” should maybe be more concise with a link to the details if people want them.

    The section on “free operads” also looks to me as though it would fit better on a page free operad.

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 24th 2010
    • (edited May 24th 2010)

    I think the material under “detailed conceptual treatment” could be condensed considerably just by recalling a few key facts, and then one could zoom in on these facts. Let’s see if I can do it here:

    (1) Set ℙ opSet^{\mathbb{P}^{op}} is the free symmetric monoidally cocomplete category on one generator.

    (2) Given symmetric monoidally cocomplete CC, DD, let SymMonCoc(C,D)SymMonCoc(C, D) denote the category of symmetric monoidal cocontinuous functors and symmetric monoidal transformations. It follows from (1) that there is an equivalence of categories

    Set ℙ op≃SymMonCoc(Set ℙ op,Set ℙ op)Set^{\mathbb{P}^{op}} \simeq SymMonCoc(Set^{\mathbb{P}^{op}}, Set^{\mathbb{P}^{op}})

    (3) The right side of this equivalence is a strict monoidal category whose monoidal product is endofunctor composition. The monoidal structure transfers across the equivalence to give monoidal category structure on Set ℙ opSet^{\mathbb{P}^{op}}. An operad is by definition a monoid in this monoidal category.

    I don’t think that’s too cryptic a summary, and one can just “zoom in” on the terms to find out what they mean. It shouldn’t be much more than a cut-and-paste job, really, since so many of the details are currently at operad.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeMay 24th 2010

    That’s a good idea, Todd.

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeFeb 18th 2011

    Todd, do you have the source of your text listed at operad – John linked there is a pdf scan which is not of best scanning quality.

    http://math.ucr.edu/home/baez/trimble/trimble_lie_operad.pdf

    By the way, Loday and Valette wrote a web draft of a new book “Algebraic operads” which I now listed in operad, link is http://www-irma.u-strasbg.fr/~loday/PAPERS/LodayVallette.pdf.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 19th 2011

    Zoran, I don’t think I have the latex file, but I do have hard copy which I can mail you if you like. (I could probably rewrite the latex file since the paper is short, and that would give me one thing (among others) to do while I’m with my in-laws all next week, starting tomorrow.) I should probably send John the missing page 16.

    • CommentRowNumber9.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 11th 2019

    Added redirects for substitution products.

    diff, v78, current

    • CommentRowNumber10.
    • CommentAuthorKeith Harbaugh
    • CommentTimeMar 4th 2021

    Added “Club” to “Related concepts”

    diff, v84, current

  11. The square brackets […] seem to stop the system from processing the math formula inside.

    Anonymous

    diff, v86, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 8th 2021
    • (edited May 8th 2021)

    I spotted one such problem in this paragraph and fixed it (the parenthetical remark there used to be enclosed in square brackets).

    If there are more issues, please give some indication where to find them (e.g. name closeby strings that one can search for)

    diff, v87, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeAug 7th 2022

    I have added publication data to:

    diff, v90, current

    • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 16th 2022

    Added reference

    diff, v91, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeAug 16th 2022

    I have added hyperlink to double functor

    diff, v92, current

    • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeAug 22nd 2022

    Do we have anywhere on the nLab a definition of operad or multicategory in terms of the binary “∘ i\circ_i” composition operations instead of the nn-ary “parallel” composition operations? I don’t see it at operad or multicategory.

    (This alternative definition is equivalent in the presence of all the other axioms, but when you start removing things, like the identity operations or the parallel associativity axiom, starting from the ∘ i\circ_i definition often gives a better result.)

    • CommentRowNumber17.
    • CommentAuthorvarkor
    • CommentTimeAug 23rd 2022

    On a related note, it would be nice to have a name for the two presentations of operad/multicategory. I think I’ve only seen names that involve ∘ i\circ_i, as you write, which is not particularly convenient.

    • CommentRowNumber18.
    • CommentAuthorMike Shulman
    • CommentTimeAug 23rd 2022

    Yeah. One possibility is the phrases I also used: “binary composition” and “parallel nn-ary composition”.

    • CommentRowNumber19.
    • CommentAuthoranuyts
    • CommentTimeFeb 7th 2023

    Operads are single-object multicategories; multicategories are colored operads.

    diff, v94, current

    • CommentRowNumber20.
    • CommentAuthorGuest
    • CommentTimeMar 12th 2023
    Swiss cheese operad is 2 colored, i.e. a slice of Swiss cheese not a 3D block. Anything been done on specifically tri-coloured?
    I have a use in mind.
    jim stasheff

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