  For a class of simplicial geometries, we construct an open cover where each
lower-dimensional simplex of codimension $p$ is covered by an intersection of
$p+1$ open sets in the open cover. We construct an injective cochain map from
the simplicial de Rham complex to the \v{C}ech-de Rham complex of the open
cover. Both the double complexes have coefficients in $L^2$, and the cochain
map we construct is between the respective domain complexes. The image of the
cochain map will be an embedding of the simplicial de Rham complex, realizing
it as a subcomplex of the \v{C}ech-de Rham complex. The simplicial de Rham
complex and the \v{C}ech-de Rham complex represent mixed-dimensional and
equidimensional coupled problems, respectively.
