group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A reduced cohomology theory is a functor
from the opposite of pointed topological spaces (CW-complexes) to $\mathbb{Z}$-graded abelian groups (“cohomology groups”), in components
and equipped with a natural isomorphism of degree +1, to be called the suspension isomorphism, of the form
such that:
(homotopy invariance) If $f_1,f_2 \colon X \longrightarrow Y$ are two morphisms of pointed topological spaces such that there is a (base point preserving) homotopy $f_1 \simeq f_2$ between them, then the induced homomorphisms of abelian groups are equal
(exactness) For $i \colon A \hookrightarrow X$ an inclusion of pointed topological spaces, with $j \colon X \longrightarrow Cone(i)$ the induced mapping cone, then this gives an exact sequence of graded abelian groups
We say $\tilde E^\bullet$ is additive if in addition
(wedge axiom) For $\{X_i\}_{i \in I}$ any set of pointed CW-complexes, then the canonical comparison morphism
is an isomorphism, from the functor applied to their wedge sum, example \ref{WedgeSumAsCoproduct}, to the product of its values on the wedge summands, .
We say $\tilde E^\bullet$ is ordinary if its value on the 0-sphere $S^0$ is concentrated in degree 0:
A homomorphism of reduced cohomology theories
is a natural transformation between the underlying functors which is compatible with the suspension isomorphisms in that all the following squares commute
(e.g. AGP 02, def. 12.1.4)
The Brown representability theorem says that for any reduced cohomology theory $\tilde E^\bullet$ there is an Omega-spectrum $E$ which represents $\tilde E^\bullet$ on pointed connected CW-complex $X$, in that
For an unreduced cohomology theory $E^\bullet$ the induced reduced cohomology is
For more see at generalized cohomology – Relation btween reduced and unreduced.
See the references at generalized (Eilenberg-Steenrod) cohomology.
Marcelo Aguilar, Samuel Gitler, Carlos Prieto, section 12 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)
Jacob Lurie, A Survey of Elliptic Cohomology - cohomology theories