On the dual nature of homology and cohomology
Standard algebraic topology defines the cup product which defines a ring structure on the cohomology of a topological space. This ring structure arises because cohomology is a contravariant functor and the pullback of the diagonal map induces the product (using the Kunneth formula for full generality, I think.)
I've always been mystified about why a dual structure, perhaps an analogous (but less conventional) "co-product", is never presented for homology. Does such a thing exist? If not, why not, and if so, is it such that the cohomology ring structure can be derived from it?
I am aware of the intersection products defined using Poincare duality, but I'm seeking a true dual to the general cup product, defined via homological algebra and valid for the all spaces with a cohomology ring.