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# Macquarie University  Department of Mathematics

## Workshop on Categorical Methods in Algebra, Geometry and Mathematical Physics

### Satellite to the StreetFest conference in honour of Ross Street's sixtieth birthday

#### July 18-21 2005, Australian National University, Canberra

Wed, 20 July:  15:15 - 15:55

##### Localic germ groupoids of inverse semigroups
###### Resende, Pedro (Instituto Superior Técnico, Lisbon)

Groupoids and inverse semigroups, which cater for more general notions of symmetry than groups, have many applications in algebra and geometry, and they are related in many ways --- in particular, there are several constructions of topological groupoids from inverse semigroups. Based on the correspondence, which I shall recall, between localic \'{e}tale groupoids and quantales that has been established in \bibref{1}, in this talk I shall study the groupoid of germs of a pseudo-group, showing that its construction can be extended to any inverse semigroup whose idempotents form a frame, yielding a localic groupoid whose spectrum is, in the case of, say, the pseudo-group of partial homeomorphisms of a Hausdorff space, the usual topological germ groupoid. The construction we provide can be carried over to an arbitrary topos (immediately yielding, for instance, a $G$-equivariant construction via an interpretation in the topos of $G$-sets), and the germ groupoid obtained is universal in the sense that, as a quantale, it is the image of an inverse semigroup by a left adjoint functor.

\begin{references}
\bibitem P. Resende, {\em Étale groupoids and their quantales}, preprint, arXiv:math/0412478.
\end{references}

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