Groupoid representations and modules over the convolution algebras

TL;DR

This paper extends the Serre-Swan theorem to etale Lie groupoids, establishing a functorial correspondence between their representations and modules over convolution algebras, generalizing classical topological results.

Contribution

It introduces a natural equivalence between the category of groupoid representations and modules over convolution algebras, broadening the scope of Serre-Swan type theorems.

Findings

Establishes a functorial correspondence between groupoid representations and modules.

Generalizes Serre-Swan theorem to etale Lie groupoids.

Provides a natural equivalence in the Morita bicategory context.

Abstract

The classical Serre-Swan's theorem defines a bijective correspondence between vector bundles and finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these results to obtain a correspondence between the category of representations of an etale Lie groupoid and the category of modules over its convolution algebra that are of finite type and of constant rank. Both of these constructions are functorially defined on the Morita bicategory of etale Lie groupoids and the given correspondence represents a natural equivalence between them.