From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5489 Path: news.gmane.org!not-for-mail From: Zinovy Diskin Newsgroups: gmane.science.mathematics.categories Subject: Re: Small is beautiful Date: Wed, 6 Jan 2010 20:10:23 -0500 Message-ID: References: Reply-To: Zinovy Diskin NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1262833985 32204 80.91.229.12 (7 Jan 2010 03:13:05 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 7 Jan 2010 03:13:05 +0000 (UTC) Cc: categories@mta.ca, pare@mathstat.dal.ca To: F William Lawvere Original-X-From: categories@mta.ca Thu Jan 07 04:12:58 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1NSinw-0006OI-Gi for gsmc-categories@m.gmane.org; Thu, 07 Jan 2010 04:12:56 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NSiOI-0000qN-MX for categories-list@mta.ca; Wed, 06 Jan 2010 22:46:26 -0400 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5489 Archived-At: In addition, a key property of semantic categories (as opposed to syntactic) is their concreteness, i.e., their objects have carriers. Commutativity with the forgetful functor is essential for results stating equivalence of syntactic and semantic constructs. Zinovy On Tue, Jan 5, 2010 at 12:31 PM, F William Lawvere w= rote: > > Bob Pare' made the excellent point that not only size but quality > is relevant. > I definitely agree with the spirit of his remarks. > > Bob happens to have used in passing the term 'syntactic'. > For clarity, the use of that term needs to be sharpened to avoid > misunderstanding. > > Actually, the term 'syntax' refers NOT to small categories such as > algebraic theories or rings, but rather to their PRESENTATION by > signatures or by polynomial generators, et cetera. The process > of presentation is an adjoint pair quite distinct from the > semantical adjoint pair: both adjoint pairs have a category of > theories or of rings in common but are otherwise quite > independent. > > In particular, syntax is NOT the adjoint of semantics. Cratylus, > Chomsky, and their 21st century followers can be refuted by > looking soberly at the actual practice of mathematics (wherein > the construction of sequences of words and of diagrams > is pursued with great care for the purpose of communication. > That syntax is only remotely dependent on the structure of the > content that is to be communicated). > > Both of the functors > > ?--------------> theories -------------->Large categories > =C2=A0 =C2=A0 =C2=A0 =C2=A0 Syntax =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0Semantics > > are needed. =C2=A0The domain category of the first can be chosen > in various useful ways: sketches or diagrams of signatures et cetera. > > Happy new year! > > Bill > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]