nLab equivariant cohomology -- table

cohomology in the presence of ∞-group $G$ ∞-action:

Borel equivariant cohomology	$\phantom{AAA}\leftarrow\phantom{AAA}$	general (Bredon) equivariant cohomology	$\phantom{AAA}\rightarrow\phantom{AAA}$	non-equivariant cohomology with homotopy fixed point coefficients
$\phantom{AA}\mathbf{H}(X_G, A)\phantom{AA}$	trivial action on coefficients $A$	$\phantom{AA}[X,A]^G\phantom{AA}$	trivial action on domain space $X$	$\phantom{AA}\mathbf{H}(X, A^G)\phantom{AA}$