From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/924 Path: news.gmane.org!not-for-mail From: street@mpce.mq.edu.au (Ross Street) Newsgroups: gmane.science.mathematics.categories Subject: Re: computation in CPS Date: Thu, 12 Nov 1998 08:21:09 +1000 Message-ID: <199811112120.IAA12919@macadam.mpce.mq.edu.au> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1241017339 28346 80.91.229.2 (29 Apr 2009 15:02:19 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:02:19 +0000 (UTC) To: categories@mta.ca Original-X-From: cat-dist Wed Nov 11 20:16:47 1998 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id TAA08282 for categories-list; Wed, 11 Nov 1998 19:08:00 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Sender: street@macadam.mpce.mq.edu.au X-MIME-Autoconverted: from quoted-printable to 8bit by mailserv.mta.ca id RAA29277 Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 60 Xref: news.gmane.org gmane.science.mathematics.categories:924 Archived-At: Dear Categories: I sent this yesterday but I notice I had mixed up the address. I'll try again. --Ross >In the omega-category In generated by the n-cube, is it possible to >find a kind of "general" formula for the (n-1)-source (target) of >the n-morphism corresponding to the interior of In ? I can do >mechanical computation in low dimension but I am not able for the >moment to imagine a formula for any dimension. In high dimension , >computations become very long. There are two algorithms: my excision of extremals and the Aitchison-Pascal triangle. The former starts with the top dimension cell and works down while the former builds up the cocycle identities recursively from dimension 0. Excision of extremals was done for simplexes on page 325 of [1] resulting in the formulas on page 330 and 331. The cubes case was done around the same time but not published. At the end of the 1980s, Ross Moore implemented this algorithm on a Mac; but things do get big quickly after the cases given in [1]. For general parity complexes, see page 330 of [2] where, even after [3], two expository mistakes remain: in the first line of the Algorithm, "largest" should be "smallest"; on the fourth line the plus signs should be unions; and on the fifth line the element w should be chosen to be not in "mu"(u)_(n+1). Aitchison presented his algorithm for simplexes and cubes at the 1987 category conference in Louvain-la-Neuve, Belgium. I explain it somewhat on page 68 of [5]; also see page 559 of [4]. The simplex case can be obtained from the cube case which has more symmetry. 1. The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987) 283-335 2. Parity complexes, Cahiers topologie et géométrie différentielle catégoriques 32 (1991) 315-343 3. Parity complexes: corrigenda, Cahiers topologie et géométrie différentielle catégoriques 35 (1994) 359-361 4. Categorical structures, Handbook of Algebra, Volume 1 (editor M. Hazewinkel; Elsevier Science, Amsterdam 1996; ISBN 0-444-82212-7) 529-577 5. Higher categories, strings, cubes and simplex equations, Applied Categorical Structures 3 (1995) 29-77 & 303 Don't be surprised that the formula was hard to imagine. John Roberts said that no amount of staring revealed the pattern. Yet, the Aitchison-Pascal triangle is the solution! I seem to remember Iain Aitchison got it first for the cubes and then, by geometry, transferred to simplexes (which is where Roberts was looking). <== Ross