From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7303 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: The Idea of Structure as Data and Conditions Date: Sat, 26 May 2012 19:48:26 -0400 (EDT) Message-ID: References: Reply-To: Michael Barr NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: dough.gmane.org 1338122634 12327 80.91.229.3 (27 May 2012 12:43:54 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 27 May 2012 12:43:54 +0000 (UTC) Cc: categories@mta.ca To: "Ellis D. Cooper" Original-X-From: majordomo@mlist.mta.ca Sun May 27 14:43:53 2012 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.80]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1SYcp3-00017d-Jo for gsmc-categories@m.gmane.org; Sun, 27 May 2012 14:43:49 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:44729) by smtpx.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1SYcnn-0008OF-Sj; Sun, 27 May 2012 09:42:31 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1SYcnn-00079a-Sd for categories-list@mlist.mta.ca; Sun, 27 May 2012 09:42:31 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7303 Archived-At: Let me point out that not every structure comes with an obvious notion of morphism. For example, if I just gave the bare-bones definition of topological space, the obvious definition of morphism would be open mappings. On complete lattices, we can have complete homomorphisms, complete sup homomorphisms and, needless to say, complete inf homomorphisms. And I have recently helped characterize the injectives in the category of partially-ordered monoids and marphisms that satisfy f(x)f(y) =< f(xy). There are Heyting algebras. Isomorphisms are always the same, so that is safe. I never understood why the founding paper in category theory was called "The general theory of natural equivalences", when they do consider more general natural transformations. Michael On Fri, 25 May 2012, Ellis D. Cooper wrote: > In the 1952 document at > http://mathdoc.emath.fr/archives-bourbaki/PDF/nbt_029.pdf the only > mathematician > "pr\'{e}sent" referenced by first name only is Sammy. > > I was permitted to audit a graduate course on category theory guided > by Sammy at Columbia University in the early 1960s. > I recall his insistence that mathematical structure is given by data > and conditions. Is that idea > implicit or explicit in Bourbaki? Has that idea been superceded? How > does it relate to the > development of algebraic theories as understood by Lawvere, Linton, > Barr-Wells, the Elephant, and so on? > > Ellis D. Cooper > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]