Date: Thu, 15 Jul 1999 10:43:56 +0200 From: Philippe Gaucher Subject: categories: Maple program for cubical categories ? Bonjour, I have calculations to do in a cubical omega-category (see works of Brown, Higgins, etc... for the definition). I have to verify some equalities. I am wondering whether there is a program (Maple, anything else) in order to simplify automatically expressions containing only the usual operators like the three families of degeneracy maps of a cubical set with connections, operations +_j, and the usual face maps. Every composition of degeneracy maps and face maps can be ordered with the degeneracy maps of the cubical sets in the first place (in a canonical order), followed by the connection maps, followed by the face maps. But I do not see a canonical way to deal with +_j (because of the interchange law for example). pg. Date: Thu, 15 Jul 1999 16:32:22 +0100 (BST) From: Ronnie Brown Subject: categories: Re: Maple program for cubical categories ? This is a very sophisticated problem, involving rewrite theory. Without the compositions, and with only one connection, there is work in 86. (with A.P.TONKS), ``Calculations with simplicial and cubical groups in AXIOM'', {\em J. Symbolic Computation}, 17 (1994) 159-179. But AXIOM is not the easiest to get working for you! I have not looked at this for a long time. The difficulties of using the interchange rule are quickly seen by considering pre crossed and crossed modules. We actually did some calculations of induced crossed modules in 92. (with C.D.WENSLEY), ``On finite induced crossed modules and the homotopy 2-type of mapping cones'', {\em Theory and Applications of Categories} 1(3) (1995) 54-71. 95. (with C.D.WENSLEY), ``Computing crossed modules induced by an inclusion of a normal subgroup, with applications to homotopy 2-types'', {\em Theory and Applications of Categories} 2 (1996) 3-16. There is more on rewriting in Heyworth-Wensley Mathematics, abstract math.CO/9907082 Logged Rewriting Procedures with Application to Identities Among Relations I thoroughly appreciate the need! Ronnie Brown On Thu, 15 Jul 1999, Philippe Gaucher wrote: > Bonjour, > > I have calculations to do in a cubical omega-category > (see works of Brown, Higgins, etc... for the definition). > I have to verify some equalities. > I am wondering whether there is a program (Maple, anything else) > in order to simplify automatically expressions containing > only the usual operators like the three families of degeneracy > maps of a cubical set with connections, operations +_j, and the > usual face maps. > > Every composition of degeneracy maps and face maps can be > ordered with the degeneracy maps of the cubical sets in the > first place (in a canonical order), followed by the connection > maps, followed by the face maps. But I do not see a canonical > way to deal with +_j (because of the interchange law for example). > > > pg. > Prof R. Brown, School of Mathematics, University of Wales, Bangor Dean St., Bangor, Gwynedd LL57 1UT, United Kingdom Tel. direct:+44 1248 382474|office: 382475 fax: +44 1248 383663 World Wide Web: home page: http://www.bangor.ac.uk/~mas010/ New article: Higher dimensional group theory Symbolic Sculpture and Mathematics: http://www.bangor.ac.uk/SculMath/ Mathematics and Knots: http://www.bangor.ac.uk/ma/CPM/exhibit/welcome.htm