  We develop the theory of mixed finite elements in terms of special inverse
systems of complexes of differential forms, defined over cellular complexes.
Inclusion of cells corresponds to pullback of forms. The theory covers for
instance composite piecewise polynomial finite elements of variable order over
polyhedral grids. Under natural algebraic and metric conditions, interpolators
and smoothers are constructed, which commute with the exterior derivative and
whose product is uniformly stable in Lebesgue spaces. As a consequence we
obtain not only eigenpair approximation for the Hodge-Laplacian in mixed form,
but also variants of Sobolev injections and translation estimates adapted to
variational discretizations.
