From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8138 Path: news.gmane.org!not-for-mail From: pjf Newsgroups: gmane.science.mathematics.categories Subject: Re: Functoriality of pullbacks of sets Date: Sat, 24 May 2014 10:17:49 -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 1400984488 11208 80.91.229.3 (25 May 2014 02:21:28 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 25 May 2014 02:21:28 +0000 (UTC) Cc: categories@mta.ca To: Vaughan Pratt Original-X-From: majordomo@mlist.mta.ca Sun May 25 04:21:22 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 1WoO3r-0001bf-TP for gsmc-categories@m.gmane.org; Sun, 25 May 2014 04:21:20 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:60524) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1WoO3X-0006xs-28; Sat, 24 May 2014 23:20:59 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1WoO3W-0002jw-UU for categories-list@mlist.mta.ca; Sat, 24 May 2014 23:20:58 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8138 Archived-At: On 2014-05-20 00:54, Vaughan Pratt wrote: > On 5/19/2014 7:03 AM, pjf wrote: >> Every category with finite limits is equivalent to a >> tau-category with a functorial choice of finite limits (and the >> construction is choice-free). > > Why merely finite? Didn't you show this for all \omega-polynomials > (i.e. less than \omega^\omega), or have I overlooked something? > > Vaughan Vaughan, \omega^\omega did play a role (40 years ago!) but not the one you describe. The full subcategory of (ZF) sets whose objects are the von Neumann ordinals has an easy tau-structure. The canonical pullbacks, for example, are those in which the order on the NW corner coincides with the order that is lexicographically induced by the two maps therefrom. (So, yes, products are strictly associative and a monic is an "inclusion" if it's order-preserving.) In section 1.4(12) of "Cats & Alligators" \omega^\omega appears as the set of objects of a full subcategory denoted _P_. METATHEOREM. A equation is true for all tau-categories iff it is true for _P_. Given a counterexample in an arbitrary tau-category to an equation in the (essentially algebraic) theory of tau-categories the proof constructs (yes, constructs) a counterexample in _P_. Best thoughts, Peter [For admin and other information see: http://www.mta.ca/~cat-dist/ ]