Subject: categories: Maple program for cubical categories ?

From: Philippe Gaucher <[email-masked]>
Date: Thu, 15 Jul 1999 10:43:56 +0200

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.

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From: Ronnie Brown <[email-masked]>
Date: Thu, 15 Jul 1999 16:32:22 +0100 (BST)

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

> way to deal with +_j (because of the interchange law for example).

Prof R. Brown, School of Mathematics, 
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