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Étale groupoids as germ groupoids and their base extensions

Part of: Semigroups Connections with other structures, applications Topological and differentiable algebraic systems

Published online by Cambridge University Press:  05 August 2010

Dmitry Matsnev  and
Pedro Resende
Dmitry Matsnev
Affiliation:
Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal (matsnev@math.ist.utl.pt; pmr@math.ist.utl.pt)
Pedro Resende
Affiliation:
Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal (matsnev@math.ist.utl.pt; pmr@math.ist.utl.pt)
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Abstract

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We introduce the notion of wide representation of an inverse semigroup and prove that with a suitably defined topology there is a space of germs of such a representation that has the structure of an étale groupoid. This gives an elegant description of Paterson's universal groupoid and of the translation groupoid of Skandalis, Tu and Yu. In addition, we characterize the inverse semigroups that arise from groupoids, leading to a precise bijection between the class of étale groupoids and the class of complete and infinitely distributive inverse monoids equipped with suitable representations, and we explain the sense in which quantales and localic groupoids carry a generalization of this correspondence.

Keywords

coarse geometrycomplete inverse semigrouplocalic groupoidtopological étale groupoidtranslation groupoiduniversal groupoid

MSC classification

Secondary: 20M18: Inverse semigroups 22A22: Topological groupoids (including differentiable and Lie groupoids) 54H10: Topological representations of algebraic systems
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 53 , Issue 3 , October 2010 , pp. 765 - 785
Copyright
Copyright © Edinburgh Mathematical Society 2010

References

1. Escardó, M. H., The regular-locally-compact coreflection of a stably locally compact locale, J. Pure Appl. Alg. 157 (2001), 41–55.Google Scholar
2. Johnstone, P. T., Stone spaces, Cambridge Studies in Advanced Mathematics, Volume 3 (Cambridge University Press, 1982).Google Scholar
3. Kumjian, A., On localizations and simple C*-algebras, Pac. J. Math. 112 (1984), 141–192.Google Scholar
4. Lawson, M. V., Inverse semigroups: the theory of partial symmetries (World Scientific, 1998).Google Scholar
5. Paterson, A. L. T., Groupoids, inverse semigroups, and their operator algebras (Birkhäuser, 1999).CrossRefGoogle Scholar
6. Renault, J., A groupoid approach to C*-algebras, Lecture Notes in Mathematics, Volume 793 (Springer, 1980).Google Scholar
7. Resende, P., Étale groupoids and their quantales, Adv. Math. 208 (2007), 147–209.CrossRefGoogle Scholar
8. Roe, J., Lectures on coarse geometry, University Lecture Series (American Mathematical Society, Providence, RI, 2003).CrossRefGoogle Scholar
9. Skandalis, G., Tu, J.-L. and Yu, G., The coarse Baum–Connes conjecture and groupoids, Topology 41 (2002), 807–834.Google Scholar