  The construction of characteristic classes via the curvature form of a
connection is one motivation for the refinement of integral cohomology by de
Rham cocycles -- known as differential cohomology. We will discuss the analog
in the case of a group action on the manifold: The definition of equivariant
characteristic forms in the Cartan model due to Nicole Berline and Mich\`ele
Vergne motivates a refinement of equivariant integral cohomology by all Cartan
cocycles. In view of this, we will also review previous definitions critically,
in particular the one given by Kiyonori Gomi.
