A VB-groupoid is a Lie groupoid equipped with a compatible linear structure.
In this paper, we describe a correspondence, up to isomorphism, between
VB-groupoids and 2-term representations up to homotopy of Lie groupoids. Under
this correspondence, the tangent bundle of a Lie groupoid G corresponds to the
"adjoint representation" of G. The value of this point of view is that the
tangent bundle is canonical, whereas the adjoint representation is not.
We define a cochain complex that is canonically associated to any
VB-groupoid. The cohomology of this complex is isomorphic to the groupoid
cohomology with values in the corresponding representations up to homotopy.
When applied to the tangent bundle of a Lie groupoid, this construction
produces a canonical complex that computes the cohomology with values in the
adjoint representation.
Finally, we give a classification of regular 2-term representations up to
homotopy. By considering the adjoint representation, we find a new
cohomological invariant associated to regular Lie groupoids.Comment: v5: Introduction is completely rewritten, many other improvements in
the exposition. v6: Implements numerous corrections and changes suggested by
referees, most notably a significant simplification of the calculations in
Appendix A.2. Final version, to appear in J. Symp. Geo
