From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9299 Path: news.gmane.org!.POSTED!not-for-mail From: Joachim Kock Newsgroups: gmane.science.mathematics.categories Subject: Re: An elementary question Date: Tue, 15 Aug 2017 23:49:17 +0200 Message-ID: References: Reply-To: Joachim Kock NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8BIT X-Trace: blaine.gmane.org 1502891492 22530 195.159.176.226 (16 Aug 2017 13:51:32 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Wed, 16 Aug 2017 13:51:32 +0000 (UTC) To: Dana Scott , categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Wed Aug 16 15:51:25 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 1dhyj7-00054P-QK for gsmc-categories@m.gmane.org; Wed, 16 Aug 2017 15:51:17 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:44143) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1dhyiZ-00079p-ND; Wed, 16 Aug 2017 10:50:43 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1dhyiD-0006HO-Hn for categories-list@mlist.mta.ca; Wed, 16 Aug 2017 10:50:21 -0300 In-reply-to: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9299 Archived-At: > 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/ ]