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University of Wales, Bangor - School of InformaticsComputational Category Theory Project |
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The development of computational category theory at Bangor arises from two streams. One is the aim of computing with groupoids, and in particular fundamental groupoids, and higher dimensional versions of them. Some background to this is given in the web article Higher Dimensional Group Theory.
A second stream is the longstanding interest of Chris Wensley in computation in groups (for example symmetric groups) and in combinatorics.
These interests came together in work on the computation of induced crossed modules in two papers (R.Brown and C.D.Wensley, `On finite induced crossed modules and the homotopy 2-type of mapping cones', Theory Appl. Categories, 3, 54-71, 1995; `Computing crossed modules induced by an inclusion of a normal subgroup, with applications to homotopy 2-types', Theory Appl. Categories, 2, 3-16, 1996.) This led Wensley to the development of a crossed module package in GAP (C.D.Wensley and M. Alp, `XMOD', GAP Share Package, 1997.)
In 1995 a successful grant application was made for an Earmarked Research Assistantship in `Identities among relations for monoids and categories', and Anne Heyworth was awarded the assistantship. Under the influence of Larry Lambe in an EPSRC Visiting Fellowship, this work developed into investigating rewriting for Kan extensions and the relations with Grobner bases, as was the subject of Anne's thesis.
BROWN, R. & HEYWORTH, A. (99.14)
Using rewriting systems to compute Kan extensions of actions of categories.
(revised version of 98.14)
xxx archive link
math.CO/9903032
The basic method of rewriting for words in a free monoid given a monoid presentation is extended to rewriting for paths in a free category given a `Kan extension presentation'. This is related to work of Carmody-Walters on the Todd-Coxeter procedure for Kan extensions, but allows for the output data to be infinite, described by a language. The result also allows rewrite methods to be applied in a greater range of situations and examples, in terms of induced actions of monoids, categories, groups or groupoids.
Ronald Brown, A. Razak Salleh
Free crossed resolutions of groups and presentations of modules of identities among relations LMS JCM, (2) 28-61.
Subj-class: Group Theory; Algebraic Topology
MSC-class: 20F05,20J05,20L10,18G50,57M07
ABSTRACT: We give formulae for a module presentation of the module of identities among relations for a presentation of a group, in terms of information on 0- and 1-combings of the Cayley graph. This is seen as a special case of extending a partial free crossed resolution of a group given a partial contracting homotopy of its universal cover.
2) with R.Brown: Using Rewrite Systems to Compute Kan Extensions of Actions of Categories, UWB Math Preprint 98.14 (submitted JSC) with: Kan Extension Program (in GAP3)
3) Rewriting as a special case of Groebner basis theory, UWB Math Preprint 98.22 (1998)
4) with A.Chandler: Groebner Bases Procedures for Testing Petri Nets, UWB Math Preprint 99.11 (1999)
5) with C.D.Wensley: Logged Rewriting with Applications to Identities Among Relations, UWB Math Preprint 99.07 (submitted LMS) (1999)
6) with B.Reinert: Reduction in ZG-modules with Applications to Identities Among Relations, UWB Math Preprint 99.09 (in preparation)
7) One-sided Noncommutative Groebner Bases with Applications to Green's Relations, UWB Math Preprint 99.10 (submitted JA) (1999)
8) Using Automata to obtain Regular Expressions for Induced Actions, UWB Math Preprint 99.19 (submitted IJAC) (1999)
9) Groebner Basis Theory for Modules, (in preparation)
10) Rewriting procedures generalise to Kan extensions of actions of categories, refereed paper, Proc. FLoC/RTA'99 (1999)
11) with R.Brown and C.D.Wensley: Logged Rewriting and Finite Derivation Type, (in preparation)
12) Logged Rewriting and Identities Among Relations for Groupoids, (in preparation)
Crossed modules: 95_03, 95_04,
XMOD share package for GAP: 97_05,
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