Operator Theory Seminar
Given a locally compact groupoid G and a locally compact abelian group bundle pA : A → G(0), an extension Σ of A is a locally compact groupoid Σ such that Σ(0) = G (0) together with maps i : A → Σ and p : Σ → G such that i is a homeomorphism onto its range, p is continuous and open, i and p restricted to G (0) are the identity. Following earlier work by Kumjian and Tu, we prove that the collection of proper isomorphism classes of compatible extensions form an abelian group. We present in detail the pushout constructions of extensions of groupoids following previous work by Kumjian. We describe how the T-groupoid of an extension is a particular example of the pushout construction. For the main examples, we specialize to extensions by 2 cocycles and prove that the pushout of such an extension is an extension by a cocycle as well. In particular, we desctibe the pushout of an extension by a normalized ech cocycle with values in a locally compact abelian group. This presentation is based on work with Alex Kumjian, Jean Renault, Aidan Sims, and Dana Williams.