From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4602 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: Bourbaki and Categories Date: Thu, 18 Sep 2008 10:31:24 -0400 (EDT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1241020051 14008 80.91.229.2 (29 Apr 2009 15:47:31 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:47:31 +0000 (UTC) To: Steve Lack , categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Sep 19 13:25:30 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 19 Sep 2008 13:25:30 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kgifd-0003RE-Fq for categories-list@mta.ca; Fri, 19 Sep 2008 13:17:25 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 72 Original-Lines: 66 Xref: news.gmane.org gmane.science.mathematics.categories:4602 Archived-At: Of course sketches are mathematical objects in their own right. Of course, the functor that assigns to each sketch the corresponding theory is not full or faithful. But the definition is precise, the notion of model is also precise, so I have no idea what, if any, content there is in the claim. Incidentally, you might with equal justice claim that triples are not mathematical objects since two distinct triples can have isomorphic categories of Eilenberg-Moore algebras. In fact there are triples (or theories) on Set that have infinitary operations, yet whose category of models is isomorphic to Set. Michael On Wed, 17 Sep 2008, Steve Lack wrote: > Dear Andrei, > > Sketches are not mathematical objects in their own right, in the same sense > that groups or spaces are. They are presentations (for theories), and have > status similar to other sorts of presentations (for groups, rings, etc.) > > Of course that is in no way meant to suggest that they are not important and > worthy of study. > > Regards, > > Steve Lack. > > > On 16/09/08 11:09 PM, "Andre.Rodin@ens.fr" wrote: > >> >> Of course, you are right about a point, I missed it! I must confess I didn't >> think about this example in precise terms. My claim is that sketch theory >> doesn't fit the structuralist (Bourbaki-Hilbertian) pattern. It hardly >> precisely fits the ancient Euclidean pattern either but there is a suggestive >> analogy, which concerns the idea that certain basic objects like point, line >> and circle *generate* the rest. >> A further claim is this: a specific reason *why* sketch theory doesn't fit the >> structuralist pattern is that in sketch theory (like in CT in general) >> isomorphisms don't have the same distinguished status. >> >> andrei >> >> >> >> >> >>> I don't know what to say about the suggestion that a circle and a line >>> make a sketch of which Euclidean plane geometry is a model. I would think >>> you would need a point too, since intersections are crucial. Maybe >>> complex projective geometry since then two lines intersect in one point >>> (unless they coincide), a line and a circle in two (unless they are >>> tangent or equal) and every pair of circles in four (ditto). Maybe the >>> exceptions could be handled in some sketch. At any rate, it wold e >>> interesting to try to sketch this in detail. At any rate, I never thought >>> about this before. >> >>> Michael >> >> >> > > >