From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8221 Path: news.gmane.org!not-for-mail From: Mike Stay Newsgroups: gmane.science.mathematics.categories Subject: Double category question Date: Mon, 21 Jul 2014 14:41:18 -0700 Message-ID: Reply-To: Mike Stay NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1406054240 18962 80.91.229.3 (22 Jul 2014 18:37:20 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 22 Jul 2014 18:37:20 +0000 (UTC) To: categories Original-X-From: majordomo@mlist.mta.ca Tue Jul 22 20:37:15 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1X9ew6-0005ZY-Vh for gsmc-categories@m.gmane.org; Tue, 22 Jul 2014 20:37:15 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:47662) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1X9evj-0002v8-TH; Tue, 22 Jul 2014 15:36:51 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1X9evi-00032L-2r for categories-list@mlist.mta.ca; Tue, 22 Jul 2014 15:36:50 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8221 Archived-At: Consider this setup: *=E2=86=92*=E2=86=92*=E2=86=92* =E2=86=93=E2=87=99=E2=86=93=E2=87=99=E2=86=93=E2=87=99=E2=86=93 *=E2=86=92*=E2=86=92*=E2=86=92* =E2=86=93=E2=87=99=E2=86=93=E2=87=99=E2=86=93=E2=87=99=E2=86=93 *=E2=86=92*=E2=86=92*=E2=86=92* =E2=86=93=E2=87=99=E2=86=93=E2=87=99=E2=86=93=E2=87=99=E2=86=93 *=E2=86=92*=E2=86=92*=E2=86=92* What kind of higher category models the case where never have a path that goes right twice in a row? There are four paths from the upper left to the lower right satisfying that condition: =E2=86=92=E2=86=93=E2=86=92=E2=86=93=E2=86=92=E2=86=93 =E2=86=92=E2=86=93=E2=86=92=E2=86=93=E2=86=93=E2=86=92 =E2=86=92=E2=86=93=E2=86=93=E2=86=92=E2=86=93=E2=86=92 =E2=86=93=E2=86=92=E2=86=93=E2=86=92=E2=86=93=E2=86=92 We can almost do this with a double category: we take the product of the points above with {0,1} and then say for horizontal neighboring points x, x' we have a single morphism (x, 0) -R-> (x', 1) and for vertical neighboring points y, y' we have two morphisms (y, 0) -D1-> (y', 0) (y, 1) -D2-> (y', 0). This way it's impossible to form the composition of two arrows going right. The squares would need to be of the form R;D2 =3D> D1;R, but the types don't match: R;D2 goes from (s,0) to (t,0) while D1;R goes from (s,0) to (t,1). Has anyone seen work on something like this? --=20 Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ]