nLab
generalized cohomology

By a generalized 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 ordinary cohomology, too. For instance there is also

• nonabelian cohomology

• sheaf cohomology

• differential cohomology

etc.

For a fully general concept of generalized cohomology, see at

• cohomology.

	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 $(-) \otimes^{\mathbb{L}} A$