  We gauge the abelian hierarchy of tensor fields in 4D by a Lie algebra. The
resulting non-abelian tensor hierarchy can be interpreted via an equivariant
chain complex. We lift this structure to N=1 superspace by constructing
superfield analogs for the tensor fields, along with covariant superfield
strengths. Next we construct Chern-Simons actions, for both the bosonic and N=1
cases, and note that the condition of gauge invariance can be presented
cohomologically. Finally, we provide an explicit realization of these
structures by dimensional reduction, for example by reducing the three-form of
eleven-dimensional supergravity into a superspace with manifest 4D, N=1
supersymmetry.
