Axioms for generalized Farrell–Tate cohomology

In Klein (Math. Ann, 319 (2001) 421– 456) we defined a variant of Farrell–Tate cohomology for a topological group G and any naive G-spectrum E by taking the homotopy cofiber of a certain norm map DGlogical andhGE→EhG. In this paper, we show how to axiomatize this theory. We then interpret the norm map as the assembly map for the homotopy fixed point functor Emaps toEhG.