group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
In cohomology of topological spaces/homotopy types, the suspension isomorphism identifies the degree-$n$ cohomology of a space $X$ with the degree-$(n+1)$ cohomology of its suspension.
For instance p. 19 in