--- Vaughan Pratt <pratt@cs.stanford.edu> wrote:
Presumably by Sup you mean what Peter Johnstone calls CSLat, complete semilattices, which is a lovely self-dual category.
Yes, indeed I did provide an equivalent definition:
[Sup denotes the category of complete lattices and sup-homomorphisms.]
According it the status of "the most awesome" however is a symptom of not yet having come to grips with the joy of Chu,
At the risk of appearing pretentious, I'd like to quote Chekhov: de gustibus, aut bene aut nihil. ;) [Incidentally, I do like Chu categories, but I will play devil's advocate here.]
a more awesome self-dual category (fully) embedding CSLat in a duality-preserving and concrete-preserving way
...but not tensor-preserving? I could just as easily say that Chu(Set,2) is (equivalent to) a lluf subcategory of Rel^2 (2 here denoting the arrow category), which is in turn (equivalent to) a full subcategory of Sup^2; the latter carries a fascinating *-autonomous structure derived from those of Sup and 2, and the composite embedding is duality-preserving (though only the first part is "concrete-preserving").
[...] which is more awesome but still not awesome to the max.
Word.
If going up only reduces the awe, then one should instead go down from CSLat for greater awe.
The trouble with (Dedekind-)infinite things is that one can argue about which way is up and which way is down. For example, both the forgetful functor Sup ---> Pos, and its left adjoint can be regarded as "embeddings" ---thus one could perversely regard complete (semi)lattices as more, not less, general than arbitrary posets.
Not only am I not a ring theorist but it's never occurred to me even to play one on the Internet.
I hope no-one would accuse me of "playing the ring theorist" on the Internet or elsewhere, merely as a result of quoting some of the subject's most celebrated theorems. [I was glad to learn that I have forgotten a smaller chunk of my undergraduate education than I would have suspected.] Cheers, Jeff. P.S. It has been pointed out to me, by a reader of this list, that the "conventional wisdom" I quoted in re the history of ideal theory is flawed (as I suspected, for no deeper reason than a profound mistrust of conventional wisdom).
[...] is commonly cited as Dedekind's original motivation for defining ideals.
Hi Jeff, in fact Kummer defined ideal numbers and proved the Fermat conjecture for regular primes before Lame' presented the fallacious argument (by some years, I think, but I can't recall just how many). There's a lot of information about this in the Edwards book named after the conjecture (and some more in his recent book on constructive algebra).