From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9297 Path: news.gmane.org!.POSTED!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Re: An elementary question Date: Mon, 14 Aug 2017 22:57:14 -0700 Message-ID: References: Reply-To: Vaughan Pratt NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 7bit X-Trace: blaine.gmane.org 1502818715 32283 195.159.176.226 (15 Aug 2017 17:38:35 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Tue, 15 Aug 2017 17:38:35 +0000 (UTC) To: Categories Original-X-From: majordomo@mlist.mta.ca Tue Aug 15 19:38:31 2017 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1dhfnR-00081q-1x for gsmc-categories@m.gmane.org; Tue, 15 Aug 2017 19:38:29 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:44007) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1dhfnH-0002Wp-LA; Tue, 15 Aug 2017 14:38:19 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1dhfmw-0001mS-99 for categories-list@mlist.mta.ca; Tue, 15 Aug 2017 14:37:58 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9297 Archived-At: > On 2017-08-13 22:55, Dana Scott wrote: >> Does (P (<) Q) have a nice categorical >> definition as a functor in the category of posets? Yes [3]. On 08/13/17 9:42 PM, Patrik Eklund wrote: > What would be the practical applications of that construction? Sequential composition, aka concatenation, aka ordinal addition [1,2]. Second diagram of Figure 1, /et seq,/ [3] answers Dana's question more generally for V-categories, with the category of preordered sets as the case V = 2. The simplified proof of Theorem 9 notwithstanding, our paper could benefit today from a more pedagogically sensitive treatment. Vaughan Pratt [1] G. Birkhoff. An extended arithmetic. Duke Mathematical Journal, 3(2), June 1937. [2] G. Birkhoff. Generalized arithmetic. Duke Mathematical Journal, 9(2), June 1942. [3] Casley, R.T., Crew, R.F., Meseguer, J., and Pratt, V.R., ``Temporal Structures'', Proc. Category Theory and Computer Science 1989, ed. D. Pitt et al, LNCS 389, 21-51, Springer-Verlag, 1989. Revised journal version in Mathematical Structures in Computer Science, Volume 1:2, 179-213, July 1991. A version missing some figures is downloadable as http://boole.stanford.edu/pub/man.pdf, the missing figures should be in the older version http://boole.stanford.edu/pub/man90.pdf. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]