Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorjcmckeown
   • CommentTimeJun 8th 2013
   • PermaLink
   Author: jcmckeown
   Format: MarkdownItexOr, how coherent must associativity be... So, 1. an interchange law $ (A \otimes B) \boxtimes (C \otimes D) = (A \boxtimes C) \otimes (B \boxtimes D) $ and 2. a fixed identity $A \otimes e = A \boxtimes e = e \otimes A = e \boxtimes A = A $ give symmetry $$ A \otimes B \sim A \boxtimes B \sim B \otimes A ;$$ 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 $E_2$? Or do we need to assume one of $\otimes,\boxtimes$ is $A_3$ or better?

   Or, how coherent must associativity be…

   So,

   1. an interchange law
      $(A \otimes B) \boxtimes (C \otimes D) = (A \boxtimes C) \otimes (B \boxtimes D)$
      and
   2. a fixed identity
      $A \otimes e = A \boxtimes e = e \otimes A = e \boxtimes A = A$

   give symmetry

   $$A \otimes B \sim A \boxtimes B \sim B \otimes A ;$$

   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 $E_2$? Or do we need to assume one of $\otimes,\boxtimes$ is $A_3$ or better?

2. • CommentRowNumber2.
   • CommentAuthorjcmckeown
   • CommentTimeJun 8th 2013
   • PermaLink
   Author: jcmckeown
   Format: MarkdownItexSilly me: of course more is needed, because $\Omega \mathbb{S}^7$ has interchange and identity, but can't be $E_2$, or else $\mathbb{S}^7$ would be $E_1$, which it certainly isn't.

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

3. • CommentRowNumber3.
   • CommentAuthorjim_stasheff
   • CommentTimeJun 9th 2013
   • PermaLink
   Author: jim_stasheff
   Format: TextI 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'

   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'
4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJun 9th 2013
   • (edited Jun 9th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> 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...

   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…

5. • CommentRowNumber5.
   • CommentAuthorjcmckeown
   • CommentTimeJun 9th 2013
   • PermaLink
   Author: jcmckeown
   Format: MarkdownItexHaving 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 $A_2$-forms on $X$, call them $\otimes,\boxtimes$ (that is, the types of $\otimes and \boxtimes$, what I called "bare type" to de-emphasize properties, are both $A\times A \to A$), 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 $A_2$ to $A_3$ — 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 $A_k$-forms, which interchange may as well be assumed to work coherently as well. Oh, interchange implies $A_3$! Ecce: $$x \otimes (y \boxtimes z) \sim (x \boxtimes e) \otimes (y \boxtimes z) \sim (x\otimes y)\boxtimes(e \otimes z) \sim (x\otimes y) \boxtimes z$$ Ah; I don't know how to get hold of James' Chicago mimeograph, but I see [you](http://nforum.mathforge.org/account/12/) wrote [the mathscinet review](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=MR&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=%20MR0123323%20&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq) for Adams' paper, [The Sphere, considered as an $H$-space, $mod p$](http://dx.doi.org/10.1093/qmath/12.1.52)... 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!

   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 $A_2$-forms on $X$, call them $\otimes,\boxtimes$ (that is, the types of $\otimes and \boxtimes$, what I called “bare type” to de-emphasize properties, are both $A\times A \to A$), 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 $A_2$ to $A_3$ — 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 $A_k$-forms, which interchange may as well be assumed to work coherently as well.

   Oh, interchange implies $A_3$! Ecce:

   $$x \otimes (y \boxtimes z) \sim (x \boxtimes e) \otimes (y \boxtimes z) \sim (x\otimes y)\boxtimes(e \otimes z) \sim (x\otimes y) \boxtimes z$$

   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 $H$-space, $mod p$… 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!

6. • CommentRowNumber6.
   • CommentAuthorjim_stasheff
   • CommentTimeJun 10th 2013
   • PermaLink
   Author: jim_stasheff
   Format: TextJames’ Chicago mimeograph?? If I knew what that referred to, I might could help.

   James’ Chicago mimeograph?? If I knew what that referred to, I might could help.
7. • CommentRowNumber7.
   • CommentAuthorjcmckeown
   • CommentTimeJun 10th 2013
   • PermaLink
   Author: jcmckeown
   Format: MarkdownItexthe "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.

   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.

8. • CommentRowNumber8.
   • CommentAuthorjim_stasheff
   • CommentTimeJun 11th 2013
   • PermaLink
   Author: jim_stasheff
   Format: Textjcm: feel free to contact me directly so I can better understand your present puzzles

   jcm: feel free to contact me directly so I can better understand your present puzzles