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The classical Serre-Swan's theorem defines a bijective correspondence betweenvector bundles and finitely generated projective modules over the algebra ofcontinuous functions on some compact Hausdorff topological space. We extendthese results to obtain a correspondence between the category ofrepresentations of an etale Lie groupoid and the category of modules over itsconvolution algebra that are of finite type and of constant rank. Both of theseconstructions are functorially defined on the Morita bicategory of etale Liegroupoids and the given correspondence represents a natural equivalence betweenthem.

Groupoid representations and modules over the convolution algebras