Locally Unitary Groupoid Crossed Products

TL;DR

This paper introduces the concept of locally unitary groupoid crossed products, establishing a correspondence between principal bundles and sheaf cohomology, and characterizing spectra of crossed products as principal bundles.

Contribution

It defines locally unitary actions and shows their spectra form principal bundles, linking crossed product spectra to principal bundle classification.

Findings

Spectrum of crossed product is a principal bundle

Isomorphism class of spectrum determines action equivalence

Every principal bundle arises as a spectrum of a locally unitary crossed product

Abstract

We define the notion of a principal S-bundle where S is a groupoid group bundle and show that there is a one-to-one correspondence between principal S-bundles and elements of a sheaf cohomology group associated to S. We also define the notion of a locally unitary action and show that the spectrum of the crossed product is a principal bundle. Furthermore, we prove that the isomorphism class of the spectrum determines the exterior equivalence class of the action and that every principal bundle can be realized as the spectrum of some locally unitary crossed product.