From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9300 Path: news.gmane.org!.POSTED!not-for-mail From: =?UTF-8?Q?Branko_Nikoli=C4=87?= Newsgroups: gmane.science.mathematics.categories Subject: Re: An elementary question Date: Thu, 17 Aug 2017 12:02:37 +1000 Message-ID: References: Reply-To: =?UTF-8?Q?Branko_Nikoli=C4=87?= NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8" X-Trace: blaine.gmane.org 1502998604 4705 195.159.176.226 (17 Aug 2017 19:36:44 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Thu, 17 Aug 2017 19:36:44 +0000 (UTC) To: Dana Scott , categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Thu Aug 17 21:36:38 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 1diQas-0000q7-EF for gsmc-categories@m.gmane.org; Thu, 17 Aug 2017 21:36:38 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:53112) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1diQaF-0006oH-T0; Thu, 17 Aug 2017 16:35:59 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1diQZs-0005C7-Ao for categories-list@mlist.mta.ca; Thu, 17 Aug 2017 16:35:36 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9300 Archived-At: Dear Dana, I'm not sure if the following construction is the one you are looking for, but it's the only categorical (in fact 2-categorical) description I could think of, and it is related to Robin Cockett's answer. If you view posets as categories (ncatlab.org/nlab/show/partial+order) then P and Q can be seen as objects of the 2-category Cat of categories, functors and natural transformations. Furthermore, instead of functors we can look at modules (aka profunctors or distributors, ncatlab.org/nlab/show/profunctor) and their morphisms, to get the bicategory Mod. The situation you described corresponds to the terminal module between Q and P (1-cell in Mod which is a terminal object in the hom-category Mod(Q,P)). The poset you obtain by taking the "modified coproduct" has the universal property of being the lax colimit of that 1-cell... A more general construction is explained here http://maths.mq.edu.au/~street/Pow.fun.pdf Best regards, Branko On 16 Aug 2017 11:50 pm, "Joachim Kock" wrote: > > Let P and Q be two posets. Define (P (<) Q) as the modified > coproduct where all the elements of P are made less than all the > elements of Q. QUESTION. Does (P (<) Q) have a nice categorical > definition as a functor in the category of posets? Hi Dana, unless I misunderstand the question, (<) is the join operation, which makes sense more generally for categories, and more generally for simplicial sets, or augmented simplicial sets. Here it is simply the cocontinuous extension (in each variable) of ordinal sum (i.e. the Day convolution tensor product of ordinal sum). (It plays an crucial role in the development of higher category theory, thanks to the discovery by Andr?? Joyal that slice and coslice can be defined as right adjoints to join with a fixed object. (These are generalised slices and coslices, with the classical notions corresponding to the cases of join with a point.) This is the construction that allows for the definition of limits and colimits in infinity-categories, and hence the starting point for generalising category theory from categories to infinity- categories.) [A. Joyal: Quasi-categories and Kan complexes, JPAA 2002] Cheers, Joachim. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]