From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8126 Path: news.gmane.org!not-for-mail From: pjf Newsgroups: gmane.science.mathematics.categories Subject: Re: Functoriality of pullbacks of sets Date: Mon, 19 May 2014 10:03:11 -0400 Message-ID: References: Reply-To: pjf NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1400572807 17050 80.91.229.3 (20 May 2014 08:00:07 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 20 May 2014 08:00:07 +0000 (UTC) Cc: categories@mta.ca To: Colin McLarty Original-X-From: majordomo@mlist.mta.ca Tue May 20 10:00:01 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 1WmXN9-0000jR-4B for gsmc-categories@m.gmane.org; Tue, 20 May 2014 01:53:35 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:59388) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1WmXMB-0008Ks-GY; Mon, 19 May 2014 20:52:35 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1WmXMB-0004Es-9s for categories-list@mlist.mta.ca; Mon, 19 May 2014 20:52:35 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8126 Archived-At: On 2014-05-18 10:59, Colin McLarty wrote: > It seems to me that Peter Freyd remarked it is easy to define pullbacks > in > ZF (maybe with with global choice?) so that pullback along one side is > functorial, but hard to make it functorial on both sides. In other > words > we can easily make base change functorial in the bases, but not easily > make > it functorial in the bases at the same time as in the total spaces. > > Can anyone direct me to a reference to that work? > > thanks, Colin The subject of "Tau-Categories" was first exposed in my 1974 mimeographed "Pamphlet," and more accessibly in my 1990 book with Andre Scedrov, "Categories, Allegories" (often called "Cats and Alligators") starting at 1.49 (p54). Every category with finite limits is equivalent to a tau-category with a functorial choice of finite limits (and the construction is choice-free). There is, indeed, a necessary asymmetry: we can have canonical pullbacks so that if both interior rectangles in the diagram .-.-. | | | .-.-. are canonical pullbacks then so is the exterior rectangle, but then it will not be the case that such holds for the rectangles in diagrams of the form .-. | | .-. | | .-. (unless, of course, the category is just a semi-lattice). By using tau-categories one can remove the use of the axiom of choice from the constructions of various representation theorems for categories. At the end of my of my 2003 Foreword to the TAC "reprinting" of "Abelian Categories" (http://www.tac.mta.ca/tac/reprints/articles/3/tr3.pdf) I remarked on how one thus gains added "functoriality" for the theorems. Best thoughts, Peter [For admin and other information see: http://www.mta.ca/~cat-dist/ ]