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.

我们将艾伦伯格与麦克莱恩提出的交叉效应（cross effects）推广至适用于同调与表示稳定性研究的范畴。具体而言，我们考虑从范畴C到带基点拓扑空间（pointed topological spaces）范畴的函子（functor），其中C为自然数范畴，或是由有限集与单射构成的范畴。我们构造了两类谱序列（spectral sequence）：一类可通过此类函子的同调（homology）计算其交叉效应的同调，另一类则可通过该函子交叉效应的同调重构其各赋值的同调。最后，我们在对称群的模2同调框架下探讨了上述交叉效应。