From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5806 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: terminology (was: bilax_monoidal_functors) Date: Fri, 14 May 2010 23:41:01 -0500 Message-ID: References: <20100514144324.D83A35C275@chase.mathstat.dal.ca> 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 1273945462 13252 80.91.229.12 (15 May 2010 17:44:22 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 15 May 2010 17:44:22 +0000 (UTC) To: Peter Selinger Original-X-From: categories@mta.ca Sat May 15 19:44:21 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 1ODLPR-00074O-42 for gsmc-categories@m.gmane.org; Sat, 15 May 2010 19:44:21 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1ODKyV-0006Js-3t for categories-list@mta.ca; Sat, 15 May 2010 14:16:31 -0300 In-Reply-To: <20100514144324.D83A35C275@chase.mathstat.dal.ca> Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5806 Archived-At: I apologize for making a "mess" by including symmetry where it was not supposed to be; I thought I was just repeating what I had heard elsewhere. Perhaps my memory was faulty, or perhaps someone else made the same error (or mess). Jeff's comment probably contributed to my confusion too. (I have read your very nice paper, but since I am only interested in a few of the notions, I didn't take the trouble to memorize all the different shadings of meaning.) Thanks for setting me straight. I am happy with "symmetric autonomous," although I guess it loses out to "compact" on the score of brevity. Mike On Fri, May 14, 2010 at 9:43 AM, Peter Selinger wrote: > Argh, Michael, you have managed to make a mess of the existing > terminology. The terminology is confusing, but it is actually > settled. While many concepts have more than one name, thankfully no > name refers to more than one concept so far (and I am working hard to > keep it that way - for example by discouraging redefinitions of > "autonomous"). Here are, for reference, the four most common notions > of (1-)categories with duals: > > (1) An "autonomous category" is a monoidal category where every object > =A0has a left dual and a right dual. Note that it is not assumed to be > =A0symmetric. There is also the notion of a "left autonomous category", > =A0where only left duals are assumed, and analogously "right autonomous > =A0category". Note that duals, where they exist, are unique up to > =A0isomorphism, so being autonomous is a property of monoidal > =A0categories, not an additional structure. > > =A0"Rigid category" is a synonym of "autonomous category", preferred by > =A0certain communities of authors. > > (2) A "pivotal category" is an autonomous category equipped with a > =A0monoidal natural isomorphism A -> A**. (A right autonomous category > =A0with such an isomorphism is automatically left autonomous too, so > =A0the right/left distinction does not apply to pivotal categories). > > =A0"Sovereign category" is a synonym of "pivotal category" used by > =A0Freyd and Yetter in one paper, but it does not seem to have caught > =A0on. It was a word play suggesting something that is even more than > =A0autonomous. > > (3) A "tortile category" is a braided pivotal (equivalently balanced > =A0autonomous) category satisfying theta* =3D theta (where theta is the > =A0twist). > > =A0"Ribbon category" is a synonym of "tortile category", preferred by > =A0certain communities of authors. > > (4) A "compact closed category" is a tortile category that is symmetric > =A0(as a balanced monoidal category), or equivalently, an autonomous > =A0symmetric monoidal category. > ... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]