From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3673 Path: news.gmane.org!not-for-mail From: John Baez Newsgroups: gmane.science.mathematics.categories Subject: more dagger problems Date: Tue, 6 Mar 2007 20:44:58 -0800 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241019448 9722 80.91.229.2 (29 Apr 2009 15:37:28 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:37:28 +0000 (UTC) To: categories Original-X-From: rrosebru@mta.ca Wed Mar 7 19:29:32 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 07 Mar 2007 19:29:32 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1HP5UF-0000Kl-5L for categories-list@mta.ca; Wed, 07 Mar 2007 19:23:59 -0400 Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 27 Original-Lines: 54 Xref: news.gmane.org gmane.science.mathematics.categories:3673 Archived-At: Juergen Koslowski wrote: >This bicategorical view also clarifies that the star operation is an >involution on 1-cells, while dagger is an involution on 2-cells. >While the name ``{1,2}-involutive bicategory'' may be adequate, >``{1,2}-involutive monoidal category'' is quite a mouthful. I call them "monoidal categories with duals". If you only have your "star", I often call them "monoidal categories with duals for objects". If you only have your "dagger", I often call them "monoidal categories with duals for morphisms". They're a special case of a fascinating notion, "k-tuply monoidal n-categories with duals", which so far only been precisely defined for low values of n and k. The "tangle hypothesis" proposes a nice topological description of the free k-tuply monoidal n-category with duals on one object. Here are some places to read about this stuff: John Baez and James Dolan, Higher-dimensional algebra and topological quantum field theory, http://arxiv.org/abs/q-alg/9503002 John Baez and Laurel Langford, Higher-dimensional algebra IV: 2-Tangles, http://arxiv.org/abs/math.QA/9811139 John Baez, Quantum computation and symmetric monoidal categories, http://golem.ph.utexas.edu/category/2006/08/quantum_computation_and_symmet.html One can also listen to lectures: Eugenia Cheng, n-categories with duals and TQFT, http://www.fields.utoronto.ca/audio/#crs-ncategories The cases that have been precisely defined include: n = 1, k = 0 categories with duals n = 1, k = 1 monoidal categories with duals n = 1, k = 2 braided monoidal categories with duals n = 1, k = 3 symmetric monoidal categories with duals n = 2, k = 0 weak 2-categories with duals n = 2, k = 1 semistrict monoidal 2-categories with duals n = 2, k = 2 semistrict braided monoidal 2-categories with duals Here "weak 2-categories" means "bicategories" and "semistrict monoidal 2-categories" means "one-object Gray-categories". For n = 1 we have up to 2 layers of duality (your "stars" and "daggers"), while for n = 2 we have up to 3. Best, jb