Van Est isomorphism for homogeneous cochains

Abstract

VB-groupoids define a special class of Lie groupoids which carry a compatible linear structure. In this paper, we show that their differentiable cohomology admits a refinement by considering the complex of cochains which are k-homogeneous on the linear fiber. Our main result is a Van Est theorem for such cochains. We also work out two applications to the general theory of representations of Lie groupoids and algebroids. The case k=1 yields a Van Est map for representations up to homotopy on 2-term graded vector bundles. Arbitrary k-homogeneous cochains on suitable VB-groupoids lead to a novel Van Est theorem for differential forms on Lie groupoids with values in a representation.

References in corpus (1)

• Linear and multiplicative 2-forms

Cited by in corpus (10)

• Morita equivalences of vector bundles
• Infinitesimal Automorphisms of VB-groupoids and algebroids
• Differential forms with values in VB-groupoids
• Deformations of Linear Lie Brackets
• Van Est differentiation and integration
• Local formulas for multiplicative forms
• Generating functions for local symplectic groupoids and non-perturbative semiclassical quantization
• Deformation Cohomology of Lie Algebroids and Morita Equivalence
• The van Est homomorphism for strict Lie 2-groups
• A Darboux classification of homogeneous Pfaffian forms on graded manifolds