Let  and  be two homotopic actions of the topological group G on the topological space X. To an object A in the G-equivariant derived category  of X relative to the action  we associate an object  of category  such that the corresponding G-equivariant compactly supported cohomologies  and  are isomorphic. When G is a Lie group and X is a subanalytic space, we prove that the G-equivariant cohomologies  and  are also isomorphic.

groups of transformations, equivariant cohomology, derived categories of sheaves.