  We hybridize the methods of finite element exterior calculus for the
Hodge-Laplace problem on differential $k$-forms in $\mathbb{R}^n$. In the cases
$k = 0$ and $k = n$, we recover well-known primal and mixed hybrid methods for
the scalar Poisson equation, while for $0 < k < n$, we obtain new hybrid finite
element methods, including methods for the vector Poisson equation in $n = 2$
and $n = 3$ dimensions. We also generalize Stenberg postprocessing from $k = n$
to arbitrary $k$, proving new superconvergence estimates. Finally, we discuss
how this hybridization framework may be extended to include nonconforming and
hybridizable discontinuous Galerkin methods.
