The purpose of this paper is to describe orbifolds in terms of (a certain kind of) groupoids. In doing so, I hope to convince you that the theory of (Lie) groupoids provides a most convenient language for developing the foundations of the theory of orbifolds. Indeed, rather than defining all kinds of structures and invariants in a seemingly ad hoc way, often in terms of local charts, one can us...

The focus of this thesis is the study of nuclearity and exactness for groupoid crossed product C∗-algebras. In particular, we present generalizations of two well-known facts from group dynamical systems and crossed products to the groupoid setting. First, we show that if G is an amenable groupoid acting on an upper-semicontinuous C∗-bundle A with nuclear section algebra A, then the associated g...

In this article we discuss some general results on the covariant Picard groupoid in the context of differential geometry and interpret the problem of lifting Lie algebra actions to line bundles in the Picard groupoid approach.

The coordinate projective line over a field is seen as a groupoid with a further ‘projection’ structure. We investigate conversely to what extent such an, abstractly given, groupoid may be coordinatized by a suitable field constructed out of the geometry.

We consider the union of two pants decompositions of the same orientable 2-dimensional surface of any genus g. Each pants decomposition corresponds to a handlebody bounded by this surface, so two pants decompositions correspond to a Heegaard splitting of a 3manifold. We introduce a groupoid acting on double pants decompositions. This groupoid is generated by two simple transformations (called f...

Multiple twinning in cubic crystals is represented geometrically by a three-dimensional fractal and algebraically by a groupoid. In this groupoid, the variant crystals are the objects, the misorientations between the variants are the operations, and the Sigma3(n) operators are the different types of operations (expressed by sets of equivalent operations). A general formula gives the number of v...

We construct an endofunctor of paths in the category of small category and show how to construct the standard homotopy invariants from it. We give a novel proof that the fundamental groupoid of a category is its associated universal groupoid.

In 1949, M. Sholander [4] showed that every medial cancellation groupoid can be embedded into a medial quasigroup. In this note we prove the dual assertion, that every medial division groupoid is a homomorphic image of a medial quasigroup.