In their classic book, Cartan and Eilenberg described a more-or-less general scheme for defining homology and cohomology theories for a number of different kinds of algebraic structure, using a general theory of augmented algebras. Later, in his doctoral dissertation, Beck showed how to use the theory of triples to derive a very different and completely general scheme for doing the same thing. Originally, it was unclear how the two theories were related, but many of these questions were eventually answered in a paper by Barr and Beck. The present paper answers the remaining such questions, most notably in the case of Lie algebras by finding a general result that takes care of all the cases at once. It also shows that it is possible to extend the Cartan-Eilenberg theory of Lie algebras from algebras that are free over the ground ring to ones that are only projective.
