From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8261 Path: news.gmane.org!not-for-mail From: Thomas Streicher Newsgroups: gmane.science.mathematics.categories Subject: cleavages and choice Date: Wed, 30 Jul 2014 17:06:44 +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 1406857082 22147 80.91.229.3 (1 Aug 2014 01:38:02 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 1 Aug 2014 01:38:02 +0000 (UTC) Cc: Categories To: Jean =?iso-8859-1?Q?B=E9nabou?= Original-X-From: majordomo@mlist.mta.ca Fri Aug 01 03:37:56 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 1XD1n9-0004IL-Iy for gsmc-categories@m.gmane.org; Fri, 01 Aug 2014 03:37:55 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:49442) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XD1mU-0004ol-Cd; Thu, 31 Jul 2014 22:37:14 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XD1mV-0007cu-1G for categories-list@mlist.mta.ca; Thu, 31 Jul 2014 22:37:15 -0300 Content-Disposition: inline In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8261 Archived-At: Dear Jean, of course, you are right when emphasizing that one need choice for classes to endow an "anonymous" fibration with a cleavage. But that applies also to catgeories with say binary products. One needs choice for classes in order to choose a product cone for every pair of objects. In many instances, however, categories come together with a choice of products and fibrations come together with a choice of a cleavage. For example Set comes with a choice of a cleavage. Fibrations arising from internal categories are even split. Many constructions on fibrations allow one to choose a cleavage given cleavages for the arguments. Do you know of any construction on fibrations which is not "cleavage preserving" in this sense? Of course, one should not require cartesian functors to preserve cleavages just as one should not require functors to preserve chosen products. Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]