Re: mystification and categorification

[Vaughan Pratt <pratt@CS.Stanford.EDU>]

>While I'm airing my confusions, can anyone tell me what
>'categorification' means? I don't know any such process; the simplest
>exanple, 'categorifying' natural numbers to get finite sets, seems to me
>rather 'remembering the finite sets and maps which gave rise to natural
>numbers by the abstraction of passing to isomorphism classes'.

A fair question.  I attended John's Coimbra lectures on this stuff in 1999
but a lot of it leaked out afterwards.  If I had to guess I'd say he was
categorifying the free monoid on one generator to make it a monoidal category,
but then how did the monoid end up as coproduct and the generator as the
final object?  One suspects some free association there---John, how *do*
you make that connection?

With regard to categorification in general, sets seem to play a central
role in at least one development of category theory.  The homobjects of
"ordinary" categories are homsets.  (In that sense I guess "ordinary" must
entail "locally small.")  2-categories are what you get if instead you let
them be homcats, suitably elaborated.

Going in the other direction, if you take homsets to be vacuous, not
in the sense that they are empty but rather that they are all the same,
then you get sets.  One more step in that direction makes everything look
the same, which may have something to do with the Maharishi Yogi hiring
category theorists for the math dept. of his university in Fairfield, Iowa.
(When I spoke last with the MY's "Minister of World Health," an MD who like
Ross Street was a classmate of mine but eight years earlier starting in 1957,
the entire conversation seemed to be largely a skirting of the minefield
of the sameness of everything, which may subconsciously have been behind my
obscure reply to Peter Freyd's posting a while back about unique existence
going back to Descartes, where I tried to one-up him by claiming it went
*much* further back.)

Categorification isn't the only way to get to 2-categories, which can be
understood instead in terms of the interchange law as a two-dimensional
associativity principle.  However John has got a lot of mileage out of
the categorification approach, which one can't begrudge in an era where
mileage and minutes are as integral to a balanced life as one's checkbook.
(Q: How many minutes in a month?  A: Depends on your plan.)

>Since in the category of sets, any nasty old infinite set satisfies
>the golden equation and many others, I have taken the liberty of
>interpreting  'nice' to mean at least 'satisfying no unexpected
>equations'.

Quite right.  I would add to this "and satisfying the expected equations."
The "nasty sets" of which Steve speaks fail to satisy such expected
equations as 2^2^X ~ X.  (The power set of a set is a Boolean algebra,
for heaven's sake.  Why on earth forget that structure prior to taking the
second exponentiation?  Set theorists seem to think that they can simply
forget structure without paying for it, but in the real world it costs
kT/2 joules per element of X to forget that structure.  If set theorists
aren't willing to pay real-world prices in their modeling, why should we
taxpayers pay them real-world salaries?  Large cardinals are a figment of
their overactive imaginations, and the solution to consistency concerns is
not to go there.)

Vaughan Pratt