Differential forms with values in VB-groupoids

TL;DR

This paper develops a Lie theory for multiplicative differential forms with values in VB-groupoids, providing a complete infinitesimal description and a differential complex, advancing the understanding of forms valued in higher representations.

Contribution

It introduces a comprehensive description of multiplicative forms with VB-groupoid values and establishes a differential complex, extending Lie theory to forms with 2-term representations up to homotopy.

Findings

Complete infinitesimal description of multiplicative forms with VB-groupoid values.

Definition of a differential complex with 1-cocycles as multiplicative forms.

Study of Morita invariance of the cohomology of this complex.

Abstract

We introduce multiplicative differential forms on Lie groupoids with values in VB-groupoids. Our main result gives a complete description of these objects in terms of infinitesimal data. By considering split VB-groupoids, we are able to present a Lie theory for differential forms on Lie groupoids with values in 2-term representations up to homotopy. We also define a differential complex whose 1-cocycles are exactly the multiplicative forms with values in VB-groupoids and study the Morita invariance of its cohomology.