From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3580 Path: news.gmane.org!not-for-mail From: "Dr. Keith G. Bowden" Newsgroups: gmane.science.mathematics.categories Subject: Re: semi direct product Date: Wed, 17 Jan 2007 13:52:12 -0000 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019390 9335 80.91.229.2 (29 Apr 2009 15:36:30 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:36:30 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Thu Jan 18 12:03:19 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 18 Jan 2007 12:03:19 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1H7ZbA-0004lg-NG for categories-list@mta.ca; Thu, 18 Jan 2007 11:54:44 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 73 Original-Lines: 66 Xref: news.gmane.org gmane.science.mathematics.categories:3580 Archived-At: Dear Bill, Your reply is slightly ambiguous. Do you mean that you call it the semidirect product by extension of the semi-direct product in group theory? Regards, Keith Bowden ----- Original Message ----- From: To: Sent: Wednesday, January 17, 2007 1:23 AM > Because Grothendieck made many constructions that > became iconic, the terminology is ambiguous. > I call this construction > "the Grothendieck semi-direct product" > because the formula for composition of these > morphisms is exactly the same as in the very special > case where I is a group. > Of course the result of the construction is a single > category "fibered" over I and every fibred category > so arises. > The original example for me (1959) was that from > Cartan-Eilenberg where I is a category of rings and > H(i) is the category of modules over i. Because > J. L. Kelley had proposed "galactic" as the analogue > at the Cat level of the traditional "local" at the level > of a space, I called such an H a "galactic cluster" . > The "fibration' terminology and the accompanying > results and definitions for descent etc were presented > by AG in Paris seminars in the very early 1960's and > can probably be accessed elecronically now. > > Best wishes > Bill > > Quoting Gaucher Philippe : > > > Dear All, > > > > Where does the Grothendieck construction come from? What is the > > original > > reference? Here is the construction. > > > > Take a functor H:I-->Cat (the category of small categories) > > > > The objects are the pairs (i,a) where a is an object of H(i). > > A morphism (i,a)-->(j,b) consists of a morphism f:i-->j of I and a > > morphism > > H(f)(a)-->b of H(j). > > > > pg. > > > > > > > > > >