By a generlized cohomology theory is usually meant a contravariant functor on a homotopy category satisfying all abstract properties of ordinary cohomology, except possibly for the dimension axiom. For more on this see at
(and for the dual concept see at generalized homology).
But there are more general generalizations of the concept of ordiary cohomology, too. For instance there is also
For a fully general concept of generalized cohomology, see at
|category theory||covariant hom||contravariant hom||tensor product|
|enriched category theory||end||end||coend|
|homotopy theory||derived hom space||cocycles||derived tensor product|