From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3577 Path: news.gmane.org!not-for-mail From: wlawvere@buffalo.edu Newsgroups: gmane.science.mathematics.categories Subject: Re: Grothendieck construction Date: Tue, 16 Jan 2007 20:23:44 -0500 Message-ID: <43796.8899649151$1241019389@news.gmane.org> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1241019388 9326 80.91.229.2 (29 Apr 2009 15:36:28 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:36:28 +0000 (UTC) To: Gaucher Philippe Original-X-From: rrosebru@mta.ca Tue Jan 16 21:23:57 2007 -0400 X-Keywords: X-UID: 70 Original-Lines: 41 Xref: news.gmane.org gmane.science.mathematics.categories:3577 Archived-At: 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. >