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Or, how coherent must associativity be…
So,
give symmetry
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 and … do we actually get that is ? Or do we need to assume one of is or better?
Silly me: of course more is needed, because has interchange and identity, but can’t be , or else would be , which it certainly isn’t.
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…
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 -forms on , call them (that is, the types of , what I called “bare type” to de-emphasize properties, are both ), 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 to — 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 -forms, which interchange may as well be assumed to work coherently as well.
Oh, interchange implies ! Ecce:
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 -space, … 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!
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.
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