Cross Effects and Stability

We consider a generalization of the cross effects of Eilenberg and MacLane to categories suitable for studying homological and representation stability. Specifically, we consider functors from a category C to the category of pointed topological spaces, where C is the category of natural numbers or the category of finite sets and injections. We construct a spectral sequence computing the homology of our cross effects from the homology of such functors, as well as a spectral sequence reconstructing the homology of the values of the functor from the homology of its cross effects. Finally, we consider these cross effects in the context of the mod 2 homology of the symmetric groups.