From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7308 Path: news.gmane.org!not-for-mail From: Andree Ehresmann Newsgroups: gmane.science.mathematics.categories Subject: Bourbaki, Ehresmann & species of structures Date: Sun, 27 May 2012 19:09:53 +0200 Message-ID: References: Reply-To: Andree Ehresmann NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset=ISO-8859-1;DelSp="Yes";format="flowed" Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1338145186 29691 80.91.229.3 (27 May 2012 18:59:46 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 27 May 2012 18:59:46 +0000 (UTC) Cc: janelg@telkomsa.net To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Sun May 27 20:59:45 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 1SYigl-0006Op-VF for gsmc-categories@m.gmane.org; Sun, 27 May 2012 20:59:40 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:44780) by smtpx.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1SYigK-0002w7-Sv; Sun, 27 May 2012 15:59:12 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1SYigL-0007tC-Pq for categories-list@mlist.mta.ca; Sun, 27 May 2012 15:59:13 -0300 In-Reply-To: Content-Disposition: inline User-Agent: Internet Messaging Program (IMP) H3 (4.1.6) Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7308 Archived-At: Dear all, Charles Ehresmann was somewhat reticent about Bourbaki's theory of =20 structures. In fact, he only participated to the first discussions on =20 the "Fascicule de r=E9sultats" since he progressively took its distances =20 with the group in the late forties. One of the reasons he gave me for =20 this withdrawal (among others) was that he thought that a good theory =20 of structures should include a theory of "local structures" which has =20 motivated a large part of his earlier works. In several papers in the early fifties, in particular in his 1952 Rio =20 de Janeiro course (reprinted in "Charles Ehresmann : Oeuvres completes =20 et comment=E9es" Part II-1) he briefly recalls the Bourbaki's =20 definition, and then proceeds to develop his own theory of local =20 structures. At this time, he did not know the notion of a category, =20 the interest of which he discovered later through one of his students. =20 It means that categories were not much discussed in France at that =20 time... The first paper where Charles uses categories is the seminal paper =20 "Gattungen von lokalen Strukturen", 1957 (reprinted in the same volume =20 II-1 of the "Oeuvres"). The aim of the paper is to define a general =20 notion of a species of structures and of a species of local structures. For that, he introduces the action of a category C on a set S, calling =20 S a species of structures over C; he proves that it corresponds to a =20 functor F from C^op to Set, and defines the associated discrete =20 fibration, which he calls a "hypermorphism category". Then he defines a species of local structures by internalizing this in =20 the category of local sets (well before internal categories were =20 known=85) and gives a "Complete Enlargement Theorem" which translates in =20 this categorical setting the construction of the species of local =20 structures associated to a pseudogroup of transformations. Though the definitions are for general categories, in the last parts =20 he restricts the categories to groupoids. (This 1957 paper has been at =20 the basis of a large part of his/our later works on internal =20 categories, sketches, completion theorems,=85) However at that time he said to me that he was not satisfied with the =20 notion of structures, and it led him to develop a more categorical =20 frame while we were in Buenos-Aires in 1958, introducing the notion of =20 "type functors" (in the paper "Cat=E9gories des foncteurs types" 1960, =20 reprinted in Volume IV-1 of the "Oeuvres" where is also given the =20 first definition of a double category). His interest of the notion of a species of structures and of =20 "covariant maps" between them has motivated the title of his 1965 book =20 "Categories et Structures" (Dunod). Yours Andree [For admin and other information see: http://www.mta.ca/~cat-dist/ ]