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Topic: Steven Cullinane is a Crank
Replies: 33   Last Post: Mar 3, 2012 2:22 PM

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 Lee Rudolph Posts: 3,143 Registered: 12/3/04
Re: Steven Cullinane is a Crank
Posted: Jul 15, 2005 6:38 AM

jdolan@math-cl-n03.math.ucr.edu (James Dolan) writes:

>in article <1vsbe.2479\$r5.563@news.indigo.ie>,
>timothy murphy <tim@birdsnest.maths.tcd.ie> wrote:
>
>|James Dolan wrote:
>|
>|> my current favorite way of thinking about the homomorphism 4! -> 3!
>|> is as the "line at infinity in the projective completion" functor
>|> from affine planes to projective lines over the field z/2.
>|
>|Er.. Do 3!, 4! mean S(3), S(4)? If so, you have only saved two
>|right-brackets.
>
>no, that is not the only thing that i've done, nor was i trying to do
>that.

What you haven't done, though, is quite finished the project of
categorifying the heck out of everything in this example. I have
often observed that anyone who can't find a canonical group structure
on a pointed 2-set just isn't trying, and now I see that a similar
condemnation should be applied to those who can't find a canonical
field structure there. So doesn't it behoove you to eliminate the
reference to "the field z/2"? ... Yes! yes! even more is true!
Unless I am quite mistaken, every pointed 4-set has a canonical
structure as an affine plane over a 2-element field (and the
the pointed 2-set of the field could be taken, canonically, to
be the natural quotient of the pointed 4-set)!!!

I'm sure that with a bit more work this could be made so elegant
that no one could understand it. (For instance, instead of starting
with this and that pointed set, one should [and this has the further
placeholders are intended to stand for positive integers by the
same notation with arbitrary {finite?} sets, thereby infuriating
Timothy Murphy a bit more] apply the "binomial coefficient" functor
X-choose-1, for X isomorphic to 2 or 4 as the case may be, wherever
possible, and then prove functioriality, universal properties, and
the whole yards-choose-9.)

Lee Rudolph

(oh! and don't forget to braid everything in sight! the homomorphism
from B4 to B3 covering that from 4! to 3! is one of nature's marvels)

Date Subject Author
7/5/05 crankbuster
7/5/05 John Ramsden
7/5/05 LarryLard
7/5/05 crankbuster
7/12/05 Steven H. Cullinane
7/12/05 crankbuster
7/12/05 John Ramsden
7/12/05 crankbuster
7/12/05 Bob Stewart
7/12/05 Bob Stewart
7/12/05 Jesse F. Hughes
7/12/05 Bob Stewart
7/12/05 Jesse F. Hughes
7/12/05 Dik T. Winter
7/12/05 Steven H. Cullinane
7/13/05 James Dolan
7/13/05 Steven H. Cullinane
7/14/05 James Dolan
7/14/05 Timothy Murphy
7/15/05 James Dolan
7/15/05 Lee Rudolph
7/15/05 crankbuster
7/15/05 Bob Stewart
7/15/05 Jesse F. Hughes
7/15/05 Jim Heckman
7/14/05 Bob Stewart
7/14/05 crankbuster
7/14/05 John Ramsden
7/13/05 Bob Stewart
12/31/05 Steven H. Cullinane
7/24/05 Steven H. Cullinane
8/4/05 Steven H. Cullinane
12/31/05 Larry Hammick
3/3/12 Frater H.