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Presentations of omega-categories by directed complexes

Part of: Categories with structure

Published online by Cambridge University Press:  09 April 2009

Sjoerd E. Crans  and
Richard Steiner
Sjoerd E. Crans
Affiliation:
School of Mathematics and Statistics University of SydneyNSW 2006, Australia
Richard Steiner
Affiliation:
Department of Mathematics University of Glasgow University Gardens, GlasgowGreat BritainG12 8QW e-mail: rjs@maths.gla.ac.uk
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Abstract

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The theory of directed complexes is extended from free ω-categories by defining presentations in which the generators are atoms and the relations are equations between molecules. Our main result relates these presentations to the more standard algebraic presentations; we also show that every ω-category has a presentation by directed complexes. The approach is similar to that used by Crans for pasting presentations.

MSC classification

Secondary: 18D05: Double categories, $2$-categories, bicategories and generalizations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 63 , Issue 1 , August 1997 , pp. 47 - 77
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Al-Agl, F. A. and Steiner, R., ‘Nerves of multiple categories’, Proc. London Math. Soc. (3) 66 (1993), 92–128.CrossRefGoogle Scholar
[2]Baez, J. C. and Dolan, J., ‘Higher dimensional algebra and topological quantum field theory’. J. Math. Phys. 36 (1995), 6073–6105.CrossRefGoogle Scholar
[3]Baues, H.-J. and Tonks, A., ‘On sum-normalized cohomology of categories, twisted homotopy pairs. and universal Toda brackets’, preprint, (Max-Planck-Institut für Mathematik, Bonn, 1995).Google Scholar
[4]Crane, L., ‘Categorical physics’, preprint hep-th 9301061, 1993.Google Scholar
[5]Crans, S. E., On combinatorial models for higher dimensional homotopies (Ph.D. Thesis, Utrecht University, 1995).Google Scholar
[8]Johnson, M., ‘The combinatorics of n-categorical pasting’, J. Pure Appl. Algebra 62 (1989), 211–225.CrossRefGoogle Scholar
[9]Kapranov, M. M. and Voevodsky, V. A., ‘Combinatorial-geometric aspects of polycategory theory: pasting schemes and higher Bruhat orders (list of results)’, Cahiers Topologie Géom. Différentielle Catégoriques 32 (1991), 11–27.Google Scholar
[10]Kapranov, M. M. and Voevodsky, V. A., ‘Braided monoidal 2-categories, 2-vector spaces and Zamolodchikov's tetrahedra equations’, J. Pure Appl. Algebra 92 (1994), 241–267.CrossRefGoogle Scholar
[11]Power, A. J., ‘An n-categorical pasting theorem’, in: Category Theory, Lecture Notes in Math. 1488 (Springer, New York, 1991) pp. 213–236.Google Scholar
[12]Power, A. J., ‘Why tricategories?’, preprint (University of Edinburgh, 1994).Google Scholar
[13]Pratt, V., ‘Modeling concurrency with geometry’, in: Principles of programming languages, proceedings of the eighteenth annual ACM symposium (ed. Wise, D.) (ACM Press, New York, 1991) pp. 311–322.Google Scholar
[14]Roberts, J. E., ‘Mathematical aspects of local cohomology’, in: Proceedings of the colloquium on operator algebras and their application to mathematical physics, Marseille (1977).Google Scholar
[15]Steiner, R., ‘The algebra of directed complexes’, Appl. Categorical Structures 1 (1993), 247–284.CrossRefGoogle Scholar
[16]Street, R., ‘Parity complexes’, Cahiers Topologie Géom. Différentielle Catégoriques 32 (1991), 315–343.Google Scholar
[17]Street, R., ‘Parity complexes: corrigenda’, Cahiers Topologie Géom. Différentielle Catégoriques 35 (1994), 359–362.Google Scholar