From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3583 Path: news.gmane.org!not-for-mail From: metere@mat.unimi.it Newsgroups: gmane.science.mathematics.categories Subject: Re: semi direct product Date: Fri, 19 Jan 2007 09:50:02 +0100 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1241019391 9345 80.91.229.2 (29 Apr 2009 15:36:31 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:36:31 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Jan 19 08:58:34 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 19 Jan 2007 08:58:34 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1H7tGK-0006qi-KZ for categories-list@mta.ca; Fri, 19 Jan 2007 08:54:32 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 76 Original-Lines: 166 Xref: news.gmane.org gmane.science.mathematics.categories:3583 Archived-At: Dear Keith, This reply to the reply is actually more ambiguous... What do you mean with "extension"? Anyway, I find it interesting, in the groupoids case, the paper: "Categorical non abelian cohomology, and the Schreier theory of groupoids", V. Blanco, M. Bullejos, E. Faro, on the arXiv as math.CT/0410202. Best regards, Beppe Metere. Quoting "Dr. Keith G. Bowden" : > 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. > > > > > > > > > > > > > > > > > > > Quoting "Dr. Keith G. Bowden" : > 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. > > > > > > > > > > > > > > > > > > > ----------------------------------------------------------------