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    • CommentRowNumber1.
    • CommentAuthorjcmckeown
    • CommentTimeJun 8th 2013

    Or, how coherent must associativity be…

    So,

    1. an interchange law
      (AB)(CD)=(AC)(BD) (A \otimes B) \boxtimes (C \otimes D) = (A \boxtimes C) \otimes (B \boxtimes D)
      and
    2. a fixed identity
      Ae=Ae=eA=eA=AA \otimes e = A \boxtimes e = e \otimes A = e \boxtimes A = A

    give symmetry

    ABABBA; A \otimes B \sim A \boxtimes B \sim B \otimes A ;

    my silly question, because I’m really not sure how much of this I’ve seen written out: If all we know before requiring 1 and 2 is the bare type of \otimes and \boxtimes… do we actually get that \otimes is E 2E_2? Or do we need to assume one of ,\otimes,\boxtimes is A 3A_3 or better?

    • CommentRowNumber2.
    • CommentAuthorjcmckeown
    • CommentTimeJun 8th 2013

    Silly me: of course more is needed, because Ω𝕊 7\Omega \mathbb{S}^7 has interchange and identity, but can’t be E 2E_2, or else 𝕊 7\mathbb{S}^7 would be E 1E_1, which it certainly isn’t.

    • CommentRowNumber3.
    • CommentAuthorjim_stasheff
    • CommentTimeJun 9th 2013
    I haven't a clue as to what ` bare type of ⊗ and ⊠…'' means
    but since you appeak to the examples of H-spaces,
    you might could find enlightening
    Frank Adams `The ten types of H-spaces'
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 9th 2013
    • (edited Jun 9th 2013)

    I haven’t a clue as to what [ it means ]

    Yeah, I think, jcmckeown, that you will get more replies if you spend just a minute to introduce your audience to your setup, your notation and your problem at hand in an at least vaguely self-contained way. We cannot read your mind! I started to guess what you might mean, but there is too much work to be done to spend much time on guessing other people’s thoughts…

    • CommentRowNumber5.
    • CommentAuthorjcmckeown
    • CommentTimeJun 9th 2013

    Having collected my spare notes (oh, Hello, Dr. Stasheff!), what I seem to be curious about at the moment is: how good is the Eckmann-Hilton argument? Given two A 2A_2-forms on XX, call them ,\otimes,\boxtimes (that is, the types of and\otimes and \boxtimes, what I called “bare type” to de-emphasize properties, are both A×AAA\times A \to A), that are too much similar (having the same identity) and compatible (interchange), then Eckmann-Hilton shows there are homotopies among them and their opposites. What I’m trying to drive at is whether [or how much] assuming more coherence improves the consequence.

    For instance, I’m now satisfied that assuming only commutativity cannot improve the A 2A_2 to A 3A_3 — being able to twist the branches of a tree doesn’t mean we can alter the separations of its branches. But the context wherein my wonder arises also suggests consideration of two interchanging A kA_k-forms, which interchange may as well be assumed to work coherently as well.

    Oh, interchange implies A 3A_3! Ecce:

    x(yz)(xe)(yz)(xy)(ez)(xy)zx \otimes (y \boxtimes z) \sim (x \boxtimes e) \otimes (y \boxtimes z) \sim (x\otimes y)\boxtimes(e \otimes z) \sim (x\otimes y) \boxtimes z

    Ah; I don’t know how to get hold of James’ Chicago mimeograph, but I see you wrote the mathscinet review for Adams’ paper, The Sphere, considered as an HH-space, modpmod p… OK, about two of the classes suggested are “commutative and not associative”, and Adams says he has examples of all ten… good to know!

    Urs, there is neither need to attempt reading my mind nor risk of offense should you decide I’m too obscure; I find as many of my difficulties in asking questions come from throwing in unneeded words (such as “bare”, here, and perhaps “type” as well) as leaving out background; but somehow Jim actually did answer at least half of my question, and rather adroitly, too!

    • CommentRowNumber6.
    • CommentAuthorjim_stasheff
    • CommentTimeJun 10th 2013
    James’ Chicago mimeograph?? If I knew what that referred to, I might could help.
    • CommentRowNumber7.
    • CommentAuthorjcmckeown
    • CommentTimeJun 10th 2013

    the “Ten Types of H-Spaces” seems to be a note by I.M.James instead of Adams, to which Adams refers in the paper I did find. But it’s OK, I’m pretty sure more of it won’t help my present puzzles.

    • CommentRowNumber8.
    • CommentAuthorjim_stasheff
    • CommentTimeJun 11th 2013
    jcm: feel free to contact me directly so I can better understand your present puzzles