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arXiv · 2107.05776

Pushouts of extensions of groupoids by bundles of abelian groups

Marius Ionescu·Alex Kumjian·Jean N. Renault·Aidan Sims·Dana P. Williams
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Abstract

We analyse extensions $Σ$ of groupoids $G$ by bundles $A$ of abelian groups. We describe a pushout construction for such extensions, and use it to describe the extension group of a given groupoid $G$ by a given bundle $A$. There is a natural action of $Σ$ on the dual of $A$, yielding a corresponding transformation groupoid. The pushout of this transformation groupoid by the natural map from the fibre product of $A$ with its dual to the Cartesian product of the dual with the circle is a twist over the transformation groupoid resulting from the action of $G$ on the dual of $A$. We prove that the full $C^*$-algebra of this twist is isomorphic to the full $C^*$-algebra of $Σ$, and that this isomorphism descends to an isomorphism of reduced algebras. We give a number of examples and applications.

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Marius Ionescu, Alex Kumjian, Jean N. Renault, Aidan Sims, Dana P. Williams. 2021-07-12. Pushouts of extensions of groupoids by bundles of abelian groups. https://arxiv.org/abs/2107.05776

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