Chain conditions, finiteness conditions, growth conditions, and other forms of finiteness, Noetherian rings and Artinian rings have been systematically studied for commutative rings and algebras since 1959. In pursuit of the deeper results of ideal theory in ordered groupoids (semigroups), it is necessary to study special classes of ordered groupoids (semigroups). Noetherian ordered groupoids (...

We use the n-globe with its skeletal filtration to define the fundamental globular ω–groupoid of a filtered space; the proofs use an analogous fundamental cubical ω–groupoid due to the author and Philip Higgins. This method also relates the construction to the fundamental crossed complex of a filtered space, and this relation allows the proof that the crossed complex associated to the free glob...

Our main aim is to associate a holonomy Lie groupoid to the connective structure of an abelian gerbe. The construction has analogies with a procedure for the holonomy Lie groupoid of a foliation, in using a locally Lie groupoid and a globalisation procedure. We show that path connections and 2–holonomy on line bundles may be formulated using the notion of a connection pair on a double category,...

Many important C∗-algebras, such as AF-algebras, Cuntz-Krieger algebras, graph algebras and foliation C∗-algebras, are the C∗-algebras of r-discrete groupoids. These C∗-algebras are often associated with inverse semigroups through the C∗-algebra of the inverse semigroup [HR90] or through a crossed product construction as in Kumjian’s localization [Kum84]. Nica [Nic94] connects groupoid C∗-algeb...

The purpose of these notes is to discuss the problem of moduli for curves of genus g ≥ 3 1 and outline the construction of the (coarse) moduli scheme of stable curves due to Gieseker. The notes are broken into 4 parts. In Section 1 we discuss the general problem of constructing a moduli “space” of curves. We will also state results about its properties, some of which will be discussed in the se...

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.