From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6407 Path: news.gmane.org!not-for-mail From: Jeff Egger Newsgroups: gmane.science.mathematics.categories Subject: Re: double 2-categories Date: Tue, 7 Dec 2010 10:35:10 -0800 (PST) Message-ID: References: Reply-To: Jeff Egger NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: dough.gmane.org 1291776131 30133 80.91.229.12 (8 Dec 2010 02:42:11 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 8 Dec 2010 02:42:11 +0000 (UTC) To: Ondrej Rypacek , categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Wed Dec 08 03:42:06 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.114]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1PQ9yo-0007zc-HD for gsmc-categories@m.gmane.org; Wed, 08 Dec 2010 03:42:06 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:42773) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PQ9yU-0006p3-RB; Tue, 07 Dec 2010 22:41:46 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PQ9yO-0006i3-0O for categories-list@mlist.mta.ca; Tue, 07 Dec 2010 22:41:40 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6407 Archived-At: Hi Ondrej, > Is there a standard reference for what could be called a double-2-category, > by which I mean a double category where the categories of horizontal and > vertical arrows are 2-categories ? Actually, it's not entirely clear to me what you mean by this (let alone whether there's a reference for it). Heard out of context, I would have guessed that "double-2-category" should mean "2-category internal to 2-Cat". This would entail, among other things: a "2-category of objects" (whose cells I shall call "objects", "vertical arrows" and "vertical discs"); a "2-category of arrows" (whose cells I shall call "horizontal arrows", "squares" and "horizontal tubes"); and, a "2-category of 2-cells" (whose cells I shall call "horizontal discs", "vertical tubes" and, um, "4-dimensional somethings"). [A horizontal tube is something whose boundary consists of two vertical discs glued to either end of a cylinder (which, in turn, consists of two squares glued together).] But this is a special case of what I am trying very hard not to call a "double-double category"---i.e., a "quadruple category". But that disagrees with what follows. > It would be a special case of a "triple category", I guess, where > there are objects, arrows in three directions, cells for each distinct > pair of the directions, and cubes surrounded by cells. So perhaps you can give some more details? Cheers, Jeff. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]