From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3948 Path: news.gmane.org!not-for-mail From: Jeff Egger Newsgroups: gmane.science.mathematics.categories Subject: Of chickens and eggs [was: is 0 prime?] Date: Tue, 2 Oct 2007 18:29:58 -0400 (EDT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241019621 11029 80.91.229.2 (29 Apr 2009 15:40:21 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:40:21 +0000 (UTC) To: categories list Original-X-From: rrosebru@mta.ca Tue Oct 2 21:42:44 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 02 Oct 2007 21:42:44 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IcsDp-0001ON-FI for categories-list@mta.ca; Tue, 02 Oct 2007 21:36:17 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 5 Original-Lines: 78 Xref: news.gmane.org gmane.science.mathematics.categories:3948 Archived-At: --- Vaughan Pratt wrote: > Presumably by Sup you mean what Peter Johnstone calls CSLat, > complete semilattices, which is a lovely self-dual category. =20 Yes, indeed I did provide an equivalent definition: > > [Sup denotes the category of complete lattices and sup-homomorphisms.= ]=20 > According it the status of "the most awesome" however is a symptom of > not yet having come to grips with the joy of Chu,=20 At the risk of appearing pretentious, I'd like to quote Chekhov: de gusti= bus, 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=20 ...but not tensor-preserving? I could just as easily say that Chu(Set,2)= =20 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;=20 the latter carries a fascinating *-autonomous structure derived from thos= e=20 of Sup and 2, and the composite embedding is duality-preserving (though o= nly=20 the first part is "concrete-preserving"). =20 > [...] which is more awesome but still not awesome to the max. =20 Word. =20 > If going up only reduces the awe, then one should instead go down from > CSLat for greater awe. =20 The trouble with (Dedekind-)infinite things is that one can argue about=20 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. =20 > Not only am I not a ring theorist but it's never occurred to me even to > play one on the Internet.=20 I hope no-one would accuse me of "playing the ring theorist" on the=20 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=20 smaller chunk of my undergraduate education than I would have suspected.]= =20 Cheers, Jeff. P.S. It has been pointed out to me, by a reader of this list, that the=20 "conventional wisdom" I quoted in re the history of ideal theory is=20 flawed (as I suspected, for no deeper reason than a profound mistrust=20 of conventional wisdom). > > [...] is commonly cited as Dedekind's original motivation=20 > > for defining ideals. >=20 > 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).