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

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

> As to the question of "why?", I have a very biased and unscientific
> answer: Sup is the most awesome category.

Oh, *there*'s the problem.  I was getting quite puzzled about all this
stuff.  Presumably by Sup you mean what Peter Johnstone calls CSLat,
complete semilattices, which is a lovely self-dual category.  (If not
ignore the following.)

According it the status of "the most awesome" however is a symptom of
not yet having come to grips with the joy of Chu, a more awesome
self-dual category (fully) embedding CSLat in a duality-preserving and
concrete-preserving way while exhibiting that duality as simply matrix
transposition, yet still not *the* most awesome.   And all that just in
Chu(Set,2).  Chu(Set,8) embeds Grp, and concretely at that, which is
more awesome but still not awesome to the max.  More awesome yet is that
you can concretely embed every category of relational structures of
total arity n in Chu(Set,2^n)---Grp fits that description on account of
the group multiplication being a ternary relation, whence Chu(Set,8)).
And so on.

If going up only reduces the awe, then one should instead go down from
CSLat for greater awe.  God and the devil command a degree of awe that
the middle class is hard pressed to match.

Not only am I not a ring theorist but it's never occurred to me even to
play one on the Internet.  On the matter of the ideals of R, it would be
very nice if they were just the endomorphisms of R but presumably that
doesn't work on the ground that not every quotient of R embeds as a
subring of R---if that's wrong then I'm really confused.

I'm not a category theorist either but I do try.  Isn't the obvious
gadget to extract from R not its lattice of ideals but its category of
quotients suitably defined?  Bill, is that what you were getting at?

Vaughan