From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4589 Path: news.gmane.org!not-for-mail From: Andre.Rodin@ens.fr Newsgroups: gmane.science.mathematics.categories Subject: Re: Bourbaki and Categories Date: Tue, 16 Sep 2008 12:27:07 +0200 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 1241020044 13954 80.91.229.2 (29 Apr 2009 15:47:24 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:47:24 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Tue Sep 16 21:18:10 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 16 Sep 2008 21:18:10 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KfkeM-0004BB-Ir for categories-list@mta.ca; Tue, 16 Sep 2008 21:12:06 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 59 Original-Lines: 109 Xref: news.gmane.org gmane.science.mathematics.categories:4589 Archived-At: When one defines, say, a group =E0 la Borbaki, i.e. structurally, it usua= lly goes without saying that the defined structure is defined up to isomorphism. T= he notion of isomorphism plays in this case the role similar to that of equa= lity in the (naive) arithmetic. In most structural constexts the distinction b= etween the "same" structure and isomorphic structures is mathematically trivial = just like the distinction between the "same" number and equal numbers. It may = be not specially discussed in this case exactly because it is very basic. The no= tion of admissible map, say, that of group homomorphism, on the contrary, requ= ires a definition, which may be non-trivial. The idea to do mathematics up to isomorphism is not Bourbaki's invention;= it goes back at least to Hilbert's "Grundlagen der Geometrie" of 1899. In th= is sense the modern axiomatic method is structuralist. In his often-quoted l= etter to Frege Hilbert explicitely says that a theory is "merely a framework" w= hile domains of their objects are multiple and transform into each other by "univocal and reversible one-one transformations". Those who trace the hi= story of mathematical structuralism back to Hilbert are quite right, in my view= . I have in mind two issues related to CT, which suggest that CT goes in a *different* direction - in spite of the fact that MacLane and many other workers in CT had (and still have) structuralist motivations. The first i= s Functorial Semantics, which brings a *category* of models, not just one m= odel up to isomorphism. From the structuralist viewpoint the presence of non-isomorphic models (i.e. non-categoricity) is a shortcoming of a given theory. From the perspective of Functorial Semantics it is a "natural" fe= ature of mathematical theories to be dealt with rather than to be remedied. The second thing I have in mind is Sketch theory. I cannot see that Hilbe= rt's basic structuralist intuition applies in this case. In my understanding t= hings work in Sketch theory more like in Euclid. Think about circle and straigh= t line as a sketch of the theory of the first four books of Euclid's "Elements".= I would particularly appreciate, Michael, your comment on this point since = I learnt a lot of Sketch theory from your works. I have also a comment about the idea to rewrite Bourbaki's "Elements" fro= m a new categorical viewpoint. Bourbaki took Euclid's "Elements" as a model for h= is work just like did Hilbert writing his "Gundlagen". In my view, this is t= his long-term Euclidean tradition of "working foundations", which is worth to= be saved and further developed, in particular in a categorical setting. I'm = less sure that Bourbaki's example should be followed in a more specific sense. Bourbaki tries to cover too much - and doesn't try to distinguish between= what belongs to foundations and what doesn't. As a result the work is too long= and not particularly usefull for (early) beginners. I realise that today's mathematics unlike mathematics of Euclid's time is vast, so it is more difficult to present its basics in a concentrated form. But consider Hilb= ert's "Grundlagen". It covers very little - actually near to nothing - of geome= try of its time. But at the same time it provided a very powerful model of how t= o do mathematics in a new way, which greatly influenced mathematics education = and mathematical research in 20th century. In my view, Euclid's "Elements" an= d Hilbert's "Grundlagen" are better examples to be followed. best, andrei le 15/09/08 12:59, Michael Barr =E0 barr@math.mcgill.ca a =E9crit : > I don't know about this. I took several courses in the late 1950s that > seem to have been influenced by the structuralist ideas (certainly > categories weren't mentioned; I never heard the word until Dave Harriso= n > arrived in 1959) and each of them started by defining an appropriate > notion of "admissible map". I do not recall any special point being ma= de > of isomorphism and I think in general it was used for what we now call = a > bimorphism (1-1 and onto) even in cases, such as topological groups, wh= en > they were not isomorphisms. > > To be sure Bourbaki was not mentioned either, but this structuralist > influence seemed strong. > > Michael >