If the objective is to generalize, then differentiation is simply the adjoint of the boundary operation, which is a homomorphism; extending the notions of close

If the objective is to generalize, then differentiation is simply the adjoint of the boundary operation, which is a homomorphism; extending the notions of closed and exact forms to general differential forms along with that the differential operator is nilpotent allows setting up de Rham cohomology, which also has analogs by multiple authors.