group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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
generalized (Eilenberg-Steenrod) cohomology.
(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
etc.
For a fully general concept of generalized cohomology, see at
homotopy | cohomology | homology | |
---|---|---|---|
$[S^n,-]$ | $[-,A]$ | $(-) \otimes A$ | |
category theory | covariant hom | contravariant hom | tensor product |
homological algebra | Ext | Ext | Tor |
enriched category theory | end | end | coend |
homotopy theory | derived hom space $\mathbb{R}Hom(S^n,-)$ | cocycles $\mathbb{R}Hom(-,A)$ | derived tensor product $(-) \otimes^{\mathbb{L}} A$ |