From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6408 Path: news.gmane.org!not-for-mail From: Ondrej Rypacek Newsgroups: gmane.science.mathematics.categories Subject: Re: double 2-categories Date: Tue, 7 Dec 2010 19:31:08 +0000 Message-ID: References: <456981.42294.qm@web110605.mail.gq1.yahoo.com> Reply-To: Ondrej Rypacek NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1291776189 30377 80.91.229.12 (8 Dec 2010 02:43:09 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 8 Dec 2010 02:43:09 +0000 (UTC) Cc: "categories@mta.ca" To: Jeff Egger Original-X-From: majordomo@mlist.mta.ca Wed Dec 08 03:43:05 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 1PQ9zk-0008Nr-8w for gsmc-categories@m.gmane.org; Wed, 08 Dec 2010 03:43:04 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:42779) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PQ9zO-0006yC-Gc; Tue, 07 Dec 2010 22:42:42 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PQ9zK-0006jS-Lj for categories-list@mlist.mta.ca; Tue, 07 Dec 2010 22:42:38 -0400 In-Reply-To: <16A709CA-BA0D-4E71-8994-F700356973D3@gmail.com> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6408 Archived-At: > Hi Jeff , > sorry, I should have been more careful . What I mean on elementary terms = is: > i) a set of objects > ii) horizontal and vertical arrows > iii) horizontal 2-cells between pairs of parallel horizontal .. arrows (iv) vertical 2-cells between paris of parallel vertical arrows (v) cells (squares) in squares of horizontal and vertical arrows (vi) cubes for a pair of squares connected at all four sides by 2-cells, horizontal at the horizontal sides, vertical at the vertical sides All of this composes in the expected way, i.e. objects, the horizontal arrows and 2-cells form a 2-category, likewise objets, vertical arrows, and vertical 2-cells. Moreover cubes compose in all three directions : horizontally along common vertical 2-cells, and vertically along common horizontal 2-cells, and in the front-to-back direction along common cells. I now believe, this is a case of what is called a 3-fold category (what I called a "triple category" before), where all arrows in one direction are identities making the double categories sharing this dimension into 2-categories. All the best, Ondrej > > On 7 Dec 2010, at 18:35, Jeff Egger wrote: > >> Hi Ondrej, >> >>> Is there a standard reference for what could be called a =A0double-2-ca= tegory, >>> by which I mean a double category where the categories of =A0horizontal= 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" shou= ld >> mean "2-category internal to 2-Cat". >> >> This would entail, among other things: >> =A0a "2-category of objects" (whose cells I shall call "objects", "verti= cal >> arrows" and "vertical discs"); >> =A0a "2-category of arrows" (whose cells I shall call "horizontal arrows= ", >> "squares" and "horizontal tubes"); and, >> =A0a "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". =A0But that >> disagrees with what follows. >> >>> It would be a special =A0case of a "triple category", I guess, where >>> there are objects, arrows in =A0three directions, cells for each distin= ct >>> pair of the directions, and cubes =A0surrounded by cells. >> >> So perhaps you can give some more details? >> >> Cheers, >> Jeff. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]