group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For a cohomology theory $E$ with cohomology groups $E(X)$ the reduced cohomology of $E$ for pointed topological spaces $X$ is on $X$ the kernel
of the map $E(X) \to E({*})$ induced from the point inclusion ${*} \to X$.