The notion of a duality between two derived functors as well as an extension
theorem for derived functors to larger categories in which they need not be
defined is introduced. These ideas are then applied to extend and study the
coext functors to an arbitrary coalgebra. A new homology theory theory is then
built therefrom and is shown to exhibit certain duality relations to the
Hochschild cohomology of certain coalgebras. Lastly, a certain exceptional type
of coalgebra is introduced and it is used to make explicit connections between
this new homology theory and the continuous cohomology of this exceptional
algebra's pro-finite dual algebra.