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.

本研究将 Eilenberg 和 MacLane 的交叉效应推广至适用于研究同调稳定性和表示稳定性的范畴。具体而言，我们考虑从范畴 C 到 pointed 拓扑空间范畴的函子，其中 C 为自然数范畴或有限集合与单射范畴。我们构建了一组谱序列，用以计算从此类函子的同调到我们所研究交叉效应的同调，以及另一组谱序列，用以从函子交叉效应的同调重建其值域的同调。最终，我们考虑这些交叉效应在对称群模 2 同调的背景之下。