  We are interested in differential forms on mixed-dimensional geometries, in
the sense of a domain containing sets of $d$-dimensional manifolds, structured
hierarchically so that each $d$-dimensional manifold is contained in the
boundary of one or more $d + 1$ dimensional manifolds.
  On any given $d$-dimensional manifold, we then consider differential
operators tangent to the manifold as well as discrete differential operators
(jumps) normal to the manifold. The combined action of these operators leads to
the notion of a semi-discrete differential operator coupling manifolds of
different dimensions. We refer to the resulting systems of equations as
mixed-dimensional, which have become a popular modeling technique for physical
applications including fractured and composite materials.
  We establish analytical tools in the mixed-dimensional setting, including
suitable inner products, differential and codifferential operators, Poincar\'e
lemma, and Poincar\'e--Friedrichs inequality. The manuscript is concluded by
defining the mixed-dimensional minimization problem corresponding to the
Hodge-Laplacian, and we show that this minimization problem is well-posed.
