Derived Functors of Graded Theories

The homotopy category of one point unions of spheres and the homotopy category of Eilenberg–MacLane spaces are examples of graded theories. Spaces yield models of these theories which can be described in terms of iterated comma categories given by functors. The associated derived functors play an important role in spectral sequences computing homotopy groups. Using cross-effect methods, the derived functors are studied and the existence of a vanishing line is proven.