From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2347 Path: news.gmane.org!not-for-mail From: Ronnie Brown Newsgroups: gmane.science.mathematics.categories Subject: Re: Function composition of natural transformations? Date: Mon, 09 Jun 2003 14:34:31 +0100 Message-ID: <3EE48CE7.D72A64B0@bangor.ac.uk> References: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018595 3702 80.91.229.2 (29 Apr 2009 15:23:15 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:23:15 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Jun 9 11:51:40 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 09 Jun 2003 11:51:40 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19PNx6-0003QN-00 for categories-list@mta.ca; Mon, 09 Jun 2003 11:48:52 -0300 X-Mailer: Mozilla 4.79 [en] (Win98; U) X-Accept-Language: en Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 38 Original-Lines: 74 Xref: news.gmane.org gmane.science.mathematics.categories:2347 Archived-At: There is another way of looking at the strict multiple globular category case, which is to use the monoidal closed structure, as ws established via the cubical case in 116. (A. AL-AGL, R. BROWN and R. STEINER), ``Multiple categories: the equivalence between a globular and cubical approach'', Advances in Mathematics 170 (2002) 71-118. This monoidal closed structure is fairly clear cubically, but is difficult to translate into globular formulae in higher dimensions. If A=END(C), where C is a multiple category (globular or cubical), so that A is one also, then the `enriched composition' is a morphism A \otimes A \to A. In low dimensions this gives left and right whiskering A_0 \times A_1 \to A_1, A_1 \times A_0 \to A_1, and there is also a function say { , }: A_1 \to A_1 \to A_2, which measures the lack of agreement of two possible definitions of compositions, and I think this is what Tom refers to in his email. In the cubical formulation, A_2 consists of `squares', and the sides of the squares are easy to interpret using whiskering. One way round the square is a.g \circ f.v and the other is f.u\circ b.g if f:a \to b, g:u \to v. In the groupoid case, ideas of this type are used in 59. (R. BROWN and N.D. GILBERT), ``Algebraic models of 3-types and automorphism structures for crossed modules'', {\em Proc. London Math. Soc.} (3) 59 (1989) 51-73. and in other papers of Nick Gilbert. The extra structure on a crossed module M (or 2-groupoid, for that matter) of a monoid morphism M \otimes M \to M allows the modelling of homotopy 3-types. However, for calculations of 3-types, crossed squares seem better, because of a Van Kampen Type theorem, not apparently available for the other structures. Ronnie Brown Tom LEINSTER wrote: > > This may be `mere' pedagogy for ordinary categories, but if you try the > same thing for 2-categories then it becomes a `genuine' issue. To put it > another way, the two different but equivalent presentations of a concept > (natural transformation) become, on categorification, significantly > different. > snip... -- Professor Emeritus R. Brown, School of Informatics, Mathematics Division, University of Wales, Bangor Dean St., Bangor, Gwynedd LL57 1UT, United Kingdom Tel. direct:+44 1248 382474|office: 382681 fax: +44 1248 361429 World Wide Web: home page: http://www.bangor.ac.uk/~mas010/ (Links to survey articles: Higher dimensional group theory Groupoids and crossed objects in algebraic topology) Centre for the Popularisation of Mathematics: http://www.cpm.informatics.bangor.ac.uk/ (reorganised site with new sculpture animations)