From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7292 Path: news.gmane.org!not-for-mail From: Colin McLarty Newsgroups: gmane.science.mathematics.categories Subject: Re: Bourbaki & category theory Date: Tue, 22 May 2012 19:06:38 -0400 Message-ID: References: <800CD7A683A74A6299D3AEC536E36256@ACERi3> Reply-To: Colin McLarty NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1337827708 23788 80.91.229.3 (24 May 2012 02:48:28 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 24 May 2012 02:48:28 +0000 (UTC) Cc: categories@mta.ca To: George Janelidze Original-X-From: majordomo@mlist.mta.ca Thu May 24 04:48:27 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 1SXO65-0005dB-9q for gsmc-categories@m.gmane.org; Thu, 24 May 2012 04:48:17 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:43706) by smtpx.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1SXO5g-0008AT-0S; Wed, 23 May 2012 23:47:52 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1SXO5e-0008Mk-9C for categories-list@mlist.mta.ca; Wed, 23 May 2012 23:47:50 -0300 In-Reply-To: <800CD7A683A74A6299D3AEC536E36256@ACERi3> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7292 Archived-At: On Tue, May 22, 2012 at 5:51 PM, George Janelidze wro= te: > Dear Colleagues, > > I don't think it is good to say that "Bourbaki had a notion of isomorphis= m > but no general notion of morphism", even in a brief message! It would not be good -- unless it was part of a longer sentence. I wrote "Prior to encountering category theory, Bourbaki had a notion of isomorphism but no general notion of morphism." The Bourbaki passage you quote was first published in 1957, at least 6 years after Bourbaki encountered category theory as shown by the letter from Weil that i quoted. best, Colin > > Bourbaki introduces morphisms in subsection 1 of section 2 of Chapter IV = of > "Theory of Sets". Removing Bourbaki's formalism, the definition can be > stated as follows: > > Let S be a class of mathematical structures of a given type, and let us > assume for simplicity that these structures have single underlying sets > (Bourbaki also makes this assumption, also just for simplicity). An eleme= nt > of S is therefore a pair (x,s), where x is a set and s a structure on x. = By > a map f : (x,s) --> (y,t) we shall mean an arbitrary map f from x to y. B= ut, > just as in category theory, a map (x,s) --> (y,t) should "remember" (x,s) > and (y,t). A class M of such maps is said to be a class of morphisms if i= t > satisfies the following conditions: > > (i) If f : (x,s) --> (y,t) and g : (y,t) --> (z,u) are in M, then so is g= f : > (x,s) --> (z,u); > > (ii) let f : (x,s) --> (y,t) be a map that is a bijection x --> y, and le= t g > : (y,t) --> (x,s) be its inverse, then f is an isomorphism if and only if > both f and g are in M. > > Here Bourbaki uses the notion of isomorphism that he (they) defined befor= e > and that is completely determined by S. That is, according to Bourbaki, a= s > soon as I know the structures I also know their isomorphisms - but I stil= l > might have flexibility in defining morphisms. How flexible it is? Well, > condition (i) says that the class of morphisms must be closed under > composition, and condition (ii) says that morphisms must nicely agree wit= h > isomorphisms. > > The readers not familiar with category theory might ask, why is it so? Wh= y > should we agree that isomorphisms are "more determined" than morphisms? > > And one needs very little category theory to answer this: Bourbaki > structures are defined as elements of sets that appear in "scales" build > using finite cartesian products and power sets (in which we see a "germ" = of > Pare theorem about toposes: no colimits!), and to define morphisms we nee= d > functoriality of scales that we don't have, since we might want to use th= e > contravariant power set functor, as e.g. for topological spaces. And this > problem disappears of course if we restrict ourselves to bijections. (In > fact even without power sets there is a problem that disappears for > bijections, but never mind). > > The citation from Weil might mean that he misunderstood (or almost > understood) exactly this... (I mean, not structures, but parts of scales > behave covariantly or contravariantly! - But apologizing to Weil, I must = say > that I have not read the full text - so maybe it is I who misunderstood > him... for instance Mac Lanes idea could be that in many cases there is t= he > "best" notion of morphism, which is correct of course). > > Talking about Bourbaki and categories, how can we not mention Charles > Ehresmann who was far ahead in understanding many aspects of category > theory, and actually used the category of sets and bijections to approach > the general concept of mathematical structure?! (It is interesting that h= is > concept of a mathematical structure is briefly mentioned in "Introduction= to > the theory of categories and functors", a book written by I. Bucur and A. > Deleanu). Surely Andree Ehresmann can tell us many interesting things abo= ut > his ideas and the reaction of the rest of Bourbaki on them. > > But let me return to Bourbaki in general. Seeing that there is certain > flexibility in choosing morphisms is great, but then not seeing that the > morphisms do not have to be maps of sets at all is absolutely crucial. So= , > yes, category theory theory was definitely invented by Mac Lane and > Eilenberg and not by Bourbaki. > > George Janelidze > > P.S. Also, as we all know, Einenberg and Mac Lane wrote "General theory o= f > natural equivalences" long before Bourbaki wrote their Chapter IV, which > itself was long before 1970. > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]