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Re: Steven Cullinane is a Crank
Posted:
Jul 15, 2005 6:38 AM
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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:
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>|> 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
advantage of overloading standard combinatorial notation in which
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)
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