From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2380 Path: news.gmane.org!not-for-mail From: Andree Ehresmann Newsgroups: gmane.science.mathematics.categories Subject: Some history on quotient categories Date: Mon, 07 Jul 2003 15:58:35 +0200 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1"; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241018615 3846 80.91.229.2 (29 Apr 2009 15:23:35 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:23:35 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Jul 7 15:18:25 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 07 Jul 2003 15:18:25 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19ZaUn-0000In-00 for categories-list@mta.ca; Mon, 07 Jul 2003 15:13:49 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 8 Original-Lines: 63 Xref: news.gmane.org gmane.science.mathematics.categories:2380 Archived-At: Here are some historical notes, in complement to Pradines' comments on=20 multiplicative graphs and on quotient categories. Charles Ehresmann introduced multiplicative graphs as an algebraic support= =20 to the notion of a "category kernel", which corresponds to the structure=20 induced on an open subset of a topological category (in the sense of=20 Charles, i.e. a category internal to Top). Such category kernels generalize= =20 group kernels (that Charles used in the second part of his thesis to define= =20 locally homogeneous spaces); we had considered them in the early sixties=20 both in his papers on differential geometry and in mine on optimization=20 problems. Then multiplicative graphs appeared as a natural tool in his=20 study on quotient categories. Charles came to quotient categories by his works on foliations and on=20 prolongations of manifolds. In his long 1962 paper on foliations (1), he=20 constructs several kinds of holonomy groupoids by a quotient process. In=20 his paper on topological categories (2), he construct categories of jets=20 (local jets and infinitesimal jets) as quotient categories of the category= =20 of local sections of a topological category. In fact he had exposed these=20 constructions much earlier in his lectures and briefly alluded to in his=20 papers on differential geometry in the late fifties. So he was naturally led to a more formal study of quotients, which he began= =20 in his paper "Structures quotient" (3), where in particular he introduces=20 the "strict quotient"; this paper (abstracted in (4)) is more easily read=20 than later papers. In a comment I have added to this paper in the "Oeuvres"= =20 (comment 170, p. 375) I mentioned other authors who have studied quotient=20 categories about the same time, in particular Dedeckerand Mersch, Higgins,= =20 Hoenke, P=FCmplun. Later on, Charles tried to develop a non-abelian cohomology for which he=20 wanted to define a natural notion of "short sequence" in Cat. For this he=20 needed to define quotients of a category or of a groupoid by a=20 sub-category. It is done in the paper (5). The main results of this paper=20 are taken back in his book "Categories et Structures" (Dunod 1965), with=20 some illustrative diagrams (it is to this book that Pradines refers). In several papers, he generalized the theory of quotient categories in the= =20 frame of internal categories (cf. "Oeuvres", Parts III and IV), . 1. "Structures feuilletees", Proc. 5th Canadian Math. Congress; reprinted= =20 in "Charles Ehresmann, Oeuvres completes et commentees" Part II-2, 563-626. 2. "Categories topologiques, III", Indig. Math. 28, 1966; reprinted in the= =20 "Oeuvres" Part II-2, 655-669. 3. "Structures quotient",.Comm. Math. Helv. 38, 1963; reprinted in=20 "Oeuvres' III-1, 143-293. 4. "Structures quotient et cegories quotient", CRAS, 256, 1963, 5031-34,=20 reprinted in "Oeuvres" III-1, 9-11. 5. "Cohomologie dans une categorie dominee", Proc. Coll. Topologie, CBRM=20 1964; reprinted in "Oeuvres" III-2, 531-590.