I'm aware to varying extents of the existence of certain decompositions of the space of $k$-forms on a compact complex or compact Riemannian manifold that split into closed, co-closed, and harmonic forms, and that the space of harmonic forms becomes isomorphic to the de Rham cohomology groups. However, these are defined differently, but there seems to be analogy here in the theorems and to some extent in the proofs; this is also suggested by the common name. Is there one? Or is there a more general way to encapsulate all this?