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-$n$ reduced cohomology of a pointed space $X$ with the degree-$(n+k)$ cohomology of its $k$-fold suspension, hence of its smash product with the $k$-sphere
In particular this serves to express cohomology in negative degree in terms of cohomology in non-negative degree of suspended spaces.
Requiring this to hold in equivariant cohomology theory not just for integer grading and spheres but also for RO(G)-grading and representation spheres leads to the concept of genuine equivariant cohomology represented by genuine G-spectra.
Textbook accounts include
In equivariant homotopy theory: