From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2585 Path: news.gmane.org!not-for-mail From: David Yetter Newsgroups: gmane.science.mathematics.categories Subject: Re: mystification and categorification Date: Fri, 5 Mar 2004 10:55:26 -0600 Message-ID: <200403051055.26794.dyetter@math.ksu.edu> References: <002a01c401ab$cd50b370$1767eb44@grassmann> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018762 4852 80.91.229.2 (29 Apr 2009 15:26:02 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:26:02 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Sun Mar 7 10:45:05 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 07 Mar 2004 10:45:05 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1AzzUR-00062i-00 for categories-list@mta.ca; Sun, 07 Mar 2004 10:42:51 -0400 User-Agent: KMail/1.5.4 In-Reply-To: <002a01c401ab$cd50b370$1767eb44@grassmann> Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 5 Original-Lines: 38 Xref: news.gmane.org gmane.science.mathematics.categories:2585 Archived-At: Categorification is a bit like quantization: it isn't a construction so much as a desideratum for a relationship between one thing and another (in the case of categorification an (n+1)-categorical structure and an n-categorical structure; in the case of quantization a quantum mechanical system and a classical mechanical system). Categorification wants to find a higher-dimensional categorical structure corresponding to a lower-dimensional one by weakening equations to natural isomorphisms and imposing new, sensible, coherence conditions. In general, for the original purpose for which it was proposed--constructions of TQFT's and models of quantum gravity--one wants the highest categorical level to have a linear structure (hence Baez wanting tensor product and a sum it distributes over, rather than cartesian product and coproduct). Specific lower-dimensional categories with structure are 'categorified' by finding a higher-dimensional category with the new structure which 'lies over' the lower dimensional one in the way an additive monoidal category lies over its Grothendieck rig. For instance any (k-linear) monoidal category with monoid of isomorphism classes M is a categorification of M, and more generally (k-linear) monoidal categories are a categorification of monoids. A simple example shows why it is not a construction: commutative monoids (as rather special categories with one object) admit two different categorifications: symmetric monoidal categories and braided monoidal categories (each regarded as a kind of bicategory with one object). There is a good argument for regarding braided monoidal categories as the 'correct' categorification: the Eckmann-Hilton theorem ('a group in GROUPS is an abelian group' or, really as the proof shows, 'a monoid in MONOIDS is a commutative monoid') 'categorifies' to: A monoidal category in MONCAT is a braided monoidal category.