Cross-effects and the classification of Taylor towers

Abstract

Let $F$ be a homotopy functor with values in the category of spectra. We show that partially stabilized cross-effects of $F$ have an action of a certain operad. For functors from based spaces to spectra, it is the Koszul dual of the little discs operad. For functors from spectra to spectra it is a desuspension of the commutative operad. It follows that the Goodwillie derivatives of $F$ are a right module over a certain “pro-operad”. For functors from spaces to spectra, the pro-operad is a resolution of the topological Lie operad. For functors from spectra to spectra, it is a resolution of the trivial operad. We show that the Taylor tower of the functor $F$ can be reconstructed from this structure on the derivatives.