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We gauge the abelian hierarchy of tensor fields in 4D by a Lie algebra. Theresulting non-abelian tensor hierarchy can be interpreted via an equivariantchain complex. We lift this structure to N=1 superspace by constructingsuperfield analogs for the tensor fields, along with covariant superfieldstrengths. Next we construct Chern-Simons actions, for both the bosonic and N=1cases, and note that the condition of gauge invariance can be presentedcohomologically. Finally, we provide an explicit realization of thesestructures by dimensional reduction, for example by reducing the three-form ofeleven-dimensional supergravity into a superspace with manifest 4D, N=1supersymmetry.

Chern-Simons Actions and Their Gaugings in 4D, N=1 Superspace