From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2393 Path: news.gmane.org!not-for-mail From: Andree Ehresmann Newsgroups: gmane.science.mathematics.categories Subject: Re: Groups vs. groupoids (Pat Donaly) Date: Wed, 16 Jul 2003 19:22:02 +0200 Message-ID: <5.1.0.14.1.20030716182606.009f0ec0@mailx.u-picardie.fr> References: <144.155b11ae.2c4454d4@aol.com> 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 1241018629 3928 80.91.229.2 (29 Apr 2009 15:23:49 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:23:49 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Jul 17 11:50:14 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 17 Jul 2003 11:50:14 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19dA32-00029I-00 for categories-list@mta.ca; Thu, 17 Jul 2003 11:47:56 -0300 In-Reply-To: <144.155b11ae.2c4454d4@aol.com> Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 21 Original-Lines: 43 Xref: news.gmane.org gmane.science.mathematics.categories:2393 Archived-At: In answer to Pat Donaly The connection between group actions and groupoids has been known and=20 extensively used for a long time. It was realized by Charles Ehresmann in=20 the early fifties. In fact Charles came to categories from groupoids, and=20 to groupoids from group actions and from pseudogroups of transformations. In particular, in his works on fibre bundles and Differential Geometry, he= =20 associated a groupoid to a pseudogroup of transformations (1), then=20 considered action of groupoids of jets as extending group actions (2). In the paper (3) he introduces topological and differentiable categories=20 (i.e., internal to Top and to Diff), in view of associating to a principal= =20 bundle H a particular topological groupoid P (called a locally trivial=20 groupoid). He then finds the locally trivial bundles associated to H as the= =20 spaces on which there is an (internal) action of this groupoid. Given a=20 topological space F with an action of a sub-group of P, he constructs such= =20 a space with fibre F by an "enlargement" process he had defined in his=20 important paper (4). These results and many others can be found in the series of papers=20 reprinted in "Charles Ehresmann : Oeuvres completes et commentees" (more=20 specially in Part I), 1980-83.. (1) Les prolongements d'une vari=E9t=E9 diff=E9rentiable, Atti IV=20 Cong. dell'Unione Mate. Italiana, Taormina 1951, reprinted in "Oeuvres",=20 Part I, pp. 207-215. (2) Introduction =E0 la th=E9orie des structures infinit=E9smales et des=20 pseudo-groupes de Lie, Actes Coll. Intern. Geom. Diff. Strasbourg, CNRS=20 1953, reprinted in "Oeuvres", Part I, pp. 217-230. (3) Categories topologiques et categories differentiables, Coll. Geom.=20 Diff. Globale, CBRM Bruxelles 1959, reprinted in "Oeuvres", Part I, pp.=20 237-250. (4) Gattungen von lokalen Strukturen, Jahres. d. Deutsches Math. 60-2,=20 1957, reprinted in "Oeuvres", Part II, pp. 125-153. Andree C. Ehresmann