From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7290 Path: news.gmane.org!not-for-mail From: "George Janelidze" Newsgroups: gmane.science.mathematics.categories Subject: Re: Bourbaki & category theory Date: Tue, 22 May 2012 23:51:28 +0200 Message-ID: References: Reply-To: "George Janelidze" NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain;format=flowed;charset="iso-8859-1";reply-type=original Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1337812355 24959 80.91.229.3 (23 May 2012 22:32:35 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 23 May 2012 22:32:35 +0000 (UTC) To: "Colin McLarty" , "Staffan Angere" , Original-X-From: majordomo@mlist.mta.ca Thu May 24 00:32:34 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 1SXK6a-0004i7-4G for gsmc-categories@m.gmane.org; Thu, 24 May 2012 00:32:32 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:43650) by smtpx.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1SXK67-0005FQ-8z; Wed, 23 May 2012 19:32:03 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1SXK68-0003mm-Jv for categories-list@mlist.mta.ca; Wed, 23 May 2012 19:32:04 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7290 Archived-At: Dear Colleagues, I don't think it is good to say that "Bourbaki had a notion of isomorphis= m=20 but no general notion of morphism", even in a brief message! Bourbaki introduces morphisms in subsection 1 of section 2 of Chapter IV = of=20 "Theory of Sets". Removing Bourbaki's formalism, the definition can be=20 stated as follows: Let S be a class of mathematical structures of a given type, and let us=20 assume for simplicity that these structures have single underlying sets=20 (Bourbaki also makes this assumption, also just for simplicity). An eleme= nt=20 of S is therefore a pair (x,s), where x is a set and s a structure on x. = By=20 a map f : (x,s) --> (y,t) we shall mean an arbitrary map f from x to y. B= ut,=20 just as in category theory, a map (x,s) --> (y,t) should "remember" (x,s)= =20 and (y,t). A class M of such maps is said to be a class of morphisms if i= t=20 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 :=20 (x,s) --> (z,u); (ii) let f : (x,s) --> (y,t) be a map that is a bijection x --> y, and le= t g=20 : (y,t) --> (x,s) be its inverse, then f is an isomorphism if and only if= =20 both f and g are in M. Here Bourbaki uses the notion of isomorphism that he (they) defined befor= e=20 and that is completely determined by S. That is, according to Bourbaki, a= s=20 soon as I know the structures I also know their isomorphisms - but I stil= l=20 might have flexibility in defining morphisms. How flexible it is? Well,=20 condition (i) says that the class of morphisms must be closed under=20 composition, and condition (ii) says that morphisms must nicely agree wit= h=20 isomorphisms. The readers not familiar with category theory might ask, why is it so? Wh= y=20 should we agree that isomorphisms are "more determined" than morphisms? And one needs very little category theory to answer this: Bourbaki=20 structures are defined as elements of sets that appear in "scales" build=20 using finite cartesian products and power sets (in which we see a "germ" = of=20 Pare theorem about toposes: no colimits!), and to define morphisms we nee= d=20 functoriality of scales that we don't have, since we might want to use th= e=20 contravariant power set functor, as e.g. for topological spaces. And this= =20 problem disappears of course if we restrict ourselves to bijections. (In=20 fact even without power sets there is a problem that disappears for=20 bijections, but never mind). The citation from Weil might mean that he misunderstood (or almost=20 understood) exactly this... (I mean, not structures, but parts of scales=20 behave covariantly or contravariantly! - But apologizing to Weil, I must = say=20 that I have not read the full text - so maybe it is I who misunderstood=20 him... for instance Mac Lanes idea could be that in many cases there is t= he=20 "best" notion of morphism, which is correct of course). Talking about Bourbaki and categories, how can we not mention Charles=20 Ehresmann who was far ahead in understanding many aspects of category=20 theory, and actually used the category of sets and bijections to approach= =20 the general concept of mathematical structure?! (It is interesting that h= is=20 concept of a mathematical structure is briefly mentioned in "Introduction= to=20 the theory of categories and functors", a book written by I. Bucur and A.= =20 Deleanu). Surely Andree Ehresmann can tell us many interesting things abo= ut=20 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=20 flexibility in choosing morphisms is great, but then not seeing that the=20 morphisms do not have to be maps of sets at all is absolutely crucial. So= ,=20 yes, category theory theory was definitely invented by Mac Lane and=20 Eilenberg and not by Bourbaki. George Janelidze P.S. Also, as we all know, Einenberg and Mac Lane wrote "General theory o= f=20 natural equivalences" long before Bourbaki wrote their Chapter IV, which=20 itself was long before 1970. -------------------------------------------------- From: "Colin McLarty" Sent: Tuesday, May 22, 2012 7:25 PM To: "Staffan Angere" Cc: Subject: categories: Re: Bourbaki & category theory > Prior to encountering category theory, Bourbaki had a notion of > isomorphism but no general notion of morphism. See this letter from. > Andre Weil to Claude Chevalley, Oct. 15, 1951: > > \begin{quotation} As you know, my honorable colleague Mac~Lane > maintains every notion of structure necessarily brings with it a > notion of homomorphism, which consists of indicating, for each of the > data that make up the structure, which ones behave covariantly and > which contravariantly [\dots] what do you think we can gain from this > kind of consideration? (quoted in Corry ~\cite[p. 380] Modern Algebra > and the Rise of Mathematical Structures}, Basel: Birkh{\"a}user > 1996.\end{quotation} > > > > On Mon, May 21, 2012 at 6:49 PM, Staffan Angere > wrote: >> Dear categorists, >> >> and also, hello everyone, since this is my first post here! I'm wonder= ing=20 >> about the connection of Bourbaki to category theory. The copy of "Theo= ry=20 >> of Sets" that I have says it's written in 1970. Yet, Dieudonn=E9 famou= sly=20 >> saiid that the theory of functors subsumed Bourbaki's theory of=20 >> structures... and, also, Bourbaki's theory of structures is very clea= rly=20 >> a theory of a type of concrete categories. On the other hand, I've see= n=20 >> claims that the categorists' use of "morphism" comes from Bourbaki. So= =20 >> who was first? Does anyone here know when Bourbaki's theory of structu= res=20 >> was really conceived? I guess this might be self-evident to anyone bo= rn=20 >> during the 1st half of the 20th century, but it has turned out to be=20 >> really hard to find out for me. >> >> Thanks in advance, >> staffan >> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]