From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8158 Path: news.gmane.org!not-for-mail From: Thomas Streicher Newsgroups: gmane.science.mathematics.categories Subject: Re: Composition of Fibrations and Quantification Date: Tue, 10 Jun 2014 09:28:41 +0200 Message-ID: References: Reply-To: Thomas Streicher NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1402410101 14837 80.91.229.3 (10 Jun 2014 14:21:41 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 10 Jun 2014 14:21:41 +0000 (UTC) Cc: Categories To: Steve Vickers Original-X-From: majordomo@mlist.mta.ca Tue Jun 10 16:21:35 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1WuMvf-0000Kz-JI for gsmc-categories@m.gmane.org; Tue, 10 Jun 2014 16:21:35 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:35289) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1WuMvF-0007k1-Eg; Tue, 10 Jun 2014 11:21:09 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1WuMvE-0002uo-7N for categories-list@mlist.mta.ca; Tue, 10 Jun 2014 11:21:08 -0300 Content-Disposition: inline In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8158 Archived-At: Adding to Steve's most appropriate comments. One also wants to have the so-called Beck-Chevalley conditions. Recall that a fibration has internal sums iff it is a bifibration where cocartesian arrows are stable under pullbacks along cartesian arrows. This property is easily seen to be preserved by composition. But P has internal products iff P^op has internal sums. However, we don't have (P \circ Q)^op = P^op \circ Q^op in particular because since the right composite doesn't exist. (If P is a fibration over BB then P^op still is a fibration over BB and not over BB^op). Still I guess one can check directly that composition preserves the property of having small products. Maybe it's even in Bart Jacob's book? Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]