Re: Ideal Theory 101 [was: is 0 prime?]

[Vaughan Pratt <pratt@cs.stanford.edu>]

wlawvere@buffalo.edu wrote:
> The awesome nature of Sup cannot be the reason why
> the Kummer functor exists, since it is merely used for=20
> recording the result. The functor is "caused" rather=20
> by an internal feature of the domain category C of=20
> commutative rings:  The category of quotient objects
> of any given R has a binary operation * that is neither
> sup nor inf even though in principle it can be=20
> expressed as a combination of limits and colimits.

Hear, hear.  As a case in point the category Div that I described,
namely the division category replacing the division lattice, has the
number lcm(m,n) (least common multiple) as pushout over the categorical
product gcd(m,n).  This pushout is not the categorical sum of m and n,
which is instead the number mn.  (It is hell dealing with sum and
product switching around like that down below.  In the upper half of
Div, namely FinSet, sum is m+n and product is mn as it is in heaven.)

> We can call it R/ab=3DR/a *R/b but how does the=20
> operation * specialize to C concretely ?

I was wondering the same thing.  I bet something good would come out of
a meeting between category theorists and ring theorists on the topic of
finding the right abstractions here---presumably a lot of the groundwork
is already in place, much as it was for UACT in 1993, although as I
recall the algebraists didn't seem in the mood at the time.

Vaughan