  This paper is concerned with the derivation and properties of differential
complexes arising from a variety of problems in differential equations, with
applications in continuum mechanics, relativity, and other fields. We present a
systematic procedure which, starting from well-understood differential
complexes such as the de Rham complex, derives new complexes and deduces the
properties of the new complexes from the old. We relate the cohomology of the
output complex to that of the input complexes and show that the new complex has
closed ranges, and, consequently, satisfies a Hodge decomposition, Poincar\'e
type inequalities, well-posed Hodge-Laplacian boundary value problems, regular
decomposition, and compactness properties on general Lipschitz domains.
