From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5492 Path: news.gmane.org!not-for-mail From: Colin McLarty Newsgroups: gmane.science.mathematics.categories Subject: Re: Small is beautiful Date: Thu, 7 Jan 2010 09:31:33 -0500 Message-ID: References: Reply-To: Colin McLarty NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1262917813 565 80.91.229.12 (8 Jan 2010 02:30:13 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 8 Jan 2010 02:30:13 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Fri Jan 08 03:30:05 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 1NT4bz-0001fP-LA for gsmc-categories@m.gmane.org; Fri, 08 Jan 2010 03:30:03 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NT4DU-0003bg-VA for categories-list@mta.ca; Thu, 07 Jan 2010 22:04:45 -0400 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5492 Archived-At: I'm not sure I understand this > 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 > =A0 =A0 =A0 =A0 Syntax =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 = =A0 =A0 =A0 =A0 =A0 =A0 =A0Semantics > > are needed. =A0The domain category of the first can be chosen > in various useful ways: sketches or diagrams of signatures et cetera. Do you mean that if we choose some kind of sketches for the domain category then theories are a reflective subcategory, more or less the 'definitionally closed' sketches? Then a presentation of a theory T would be (up to isomorphism) any unit arrow of the adjunction with T as codomain? best, Colin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]