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February 1, 2008, 11 pages, UWB Math Preprint 04.15

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An extended account of a short presentation with this title given at the Minneapolis IMA Workshop on n--categories: foundations and applications, June 7--18, 2004, free downloads at: http://arxiv.org/PS_cache/math/pdf/0407/0407275v2.pdf historical context for the current theory.========================================== References [1] M. Alp and C. D. Wensley, XMod, Crossed modules and Cat1-groups: a GAP4 package, (2004) (http://www.maths.bangor.ac.uk/chda/) [2] R. Brown, Elements of Modern Topology, McGraw Hill, Maidenhead, 1968. second edition as Topology: a geometric account of general topology, homotopy types, and the fundamental groupoid, Ellis Horwood, Chichester (1988) 460 pp. [3] R. Brown, Higher dimensional group theory,http://www.bangor.ac.uk/~mas010/hdaweb2.htm [4] R. Brown,Crossed complexes and homotopy groupoids as non commutative tools for higherdimensional local-to-global problems, Proceedings of the Fields Institute Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories, September 23-28, 2002, Contemp. Math. (2004). (to appear), UWB Math Preprint 02.26.pdf (30 pp.) [5] R. Brown and P. J. Higgins, On the connection between the second relative homotopy groups of some related spaces, Proc.London Math. Soc., (3) 36 (1978) 193--212. [6] R. Brown and R. Sivera,"Nonabelian algebraic topology", Part I is downloadable from (http://www.bangor.ac.uk/~mas010/nonab-a-t.html) [7] R. Brown and C. B. Spencer, Double groupoids and crossed modules, Cahiers Top. Geom.Diff., 17 (1976) 343--362. [8] R. Brown and C. D.Wensley,Computation and homotopical applications of induced crossedmodules, J. Symbolic Computation, 35 (2003) 59--72. [9] The GAP Group, 2004, GAP.Groups, Algorithms, and Programming, version 4.4 , Technical report, (http://www.gap--system.org) [10] A. Grothendieck, Pursuing Stacks, 600p, 1983, distributed from Bangor. Now being edited by G. Maltsiniotis for the SMF. [11] P. J. Higgins, 1971, Categories and Groupoids, Van Nostrand, New York. Reprint Series, Theory and Appl. Categories. [12] V. Sharko, 1993, Functions on manifolds: algebraic and topological aspects, number 131 in Translations of Mathematical Monographs, American Mathematical Society. Classification: msc:55-01, msc:55N20, msc:55N40, msc:55N99, msc:55N15, msc:55N30, msc:18-00, msc:11E72, msc:11F23, msc:57N65, msc:57R19 http://arxiv.org/PS_cache/math/pdf/0407/0407275v2.pdf

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