From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6392 Path: news.gmane.org!not-for-mail From: Alan Jeffrey Newsgroups: gmane.science.mathematics.categories Subject: Re: Is "braided monoidal" a conservative extension of "lax braided monoidal"? Date: Tue, 30 Nov 2010 17:43:22 -0600 Message-ID: References: Reply-To: Alan Jeffrey NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="ISO-8859-1"; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1291208824 32540 80.91.229.12 (1 Dec 2010 13:07:04 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 1 Dec 2010 13:07:04 +0000 (UTC) Cc: To: Ross Street Original-X-From: majordomo@mlist.mta.ca Wed Dec 01 14:06:58 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 1PNmOX-0000Zt-NI for gsmc-categories@m.gmane.org; Wed, 01 Dec 2010 14:06:49 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:52344) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PNmOB-0003XF-1O; Wed, 01 Dec 2010 09:06:27 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PNmO7-0002Jt-0C for categories-list@mlist.mta.ca; Wed, 01 Dec 2010 09:06:23 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6392 Archived-At: Indeed, a categorification of Garside's embedding of the positive braid=20 monoid into the braid group is exactly what I'm after. Unfortunately,=20 the nearest published result I've been able to find is: Left-Garside categories, self-distributivity, and braids Patrick Dehornoy http://ambp.cedram.org/item?id=3DAMBP_2009__16_2_189_0 which shows this result for "the positive braid category", which has=20 natural numbers as objects, and braids as morphisms (and hence there are=20 only morphisms m --> n when m =3D n). The case of a freely generated lax= =20 braided monoidal category isn't covered, although it may follow similarly= . There's a couple of related works: Garside categories, periodic loops and cyclic sets David Bessis http://arxiv.org/abs/math/0610778 Garside and locally Garside categories Fran=E7ois Digne and Jean Michel http://arxiv.org/abs/math/0612652 but they don't appear to have exactly the categorification of Garside=20 either. A. On 11/29/2010 04:24 PM, Ross Street wrote: > Dear Alan > The positive braid monoid is a Garside monoid and it embeds > in the braid group. > (It is not generally true that a monoid embeds in its group of > fractions. > There is a literature on Garside monoids.) > I think this may be what you need. > Ross > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]