From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5783 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: bilax_monoidal_functors?= Date: Mon, 10 May 2010 20:04:46 -0500 Message-ID: References: <4BE81F26.4020903@dm.uba.ar> Reply-To: Michael Shulman 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 1273625552 15896 80.91.229.12 (12 May 2010 00:52:32 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 12 May 2010 00:52:32 +0000 (UTC) To: John Baez Original-X-From: categories@mta.ca Wed May 12 02:52:31 2010 connect(): No such file or directory Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OC0Ba-000238-PB for gsmc-categories@m.gmane.org; Wed, 12 May 2010 02:52:30 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OBzd8-0007RP-D2 for categories-list@mta.ca; Tue, 11 May 2010 21:16:54 -0300 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5783 Archived-At: I think it is the least confusing for everyone if when "foo"s start being decorated with numbers, a "1-foo" is the same thing as what an unadorned "foo" used to be. So I definitely have to agree that an ordinary braided monoidal category should be called "1-braided" if the naming scheme is going to go by decorating "braided" with numbers. On the other hand, occasionally it seems to happen that after "foo"s have been studied for a while, someone introduces a categorified "foo" and calls it a "bar," and then later someone else comes along and categorifies again but now starts introducing numbers with "2-bar," "3-bar," and so on. So what really should have been called a "2-foo" is called a "bar," what really should have been called a "3-foo" is called a "2-bar," and so on with the numbers all off by one. As John points out, the use of "braided =3D 1-braided" and then "2-braided," etc. could be viewed this way, with "monoidal" as the basic "foo" that we should have started numbering at. (One other example of this that comes to mind is the original use of "stack" to mean essentially "2-sheaf," leading to "2-stack" for something that is really a 3-categorical object, and so on. Fortunately this particular trend seems to be reversing somewhat.) However, in the case at hand, it seems to me that there is also an advantage to the term "braided" over "doubly monoidal." To give a category a braided monoidal structure may be *equivalent* to giving it two interchanging monoidal structures, but that's only true because in the latter case, the interchange law forces the two monoidal structures to be essentially the same. In practice, I find that I very rarely think about a braided monoidal category as if it were equipped with two monoidal structures; rather I think of it as having one monoidal structure together with an extra structure called a "braiding." So there are arguments on both sides of this issue, and as John says probably neither usage will create any confusion. Mike On Mon, May 10, 2010 at 1:16 PM, John Baez wrote: > Eduardo wrote: > > >> Andre points out: >> >> "To call a monoidal category 1-braided is kind of confusing because ther= e >> is no commutation structure on a general monoidal category. A monoidal >> category is 0-braided." >> >> Being an outsider, with no previous neither usage or opinion on this >> terminology beyond just monoidal and/or tensor category, this seems to m= e >> definitive, and more than enough to settle the question. > > > I'm glad that's enough to convince you that Michael Batanin's terminology > "monoidal =3D 1-braided" is inferior to Andre's "monoidal =3D 0-braided". > > But I think "braided =3D doubly monoidal" is even better. =A0After all, a > monoidal category has one tensor product; a braided monoidal category has > two compatible tensor products, and a symmetric monoidal category has thr= ee. > > > But I will not lose sleep if Andre uses "k-braided" as a synonym for > "(k+1)-tuply monoidal". =A0I don't see it causing any confusion. I just t= hink > it will create more +1's in various formulas. =A0E.g.: the classifying sp= ace > of a k-braided n-category is a (k+1)-fold loop space. > > Best, > jb > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]