Let be a homotopy functor with values in the category of spectra. We show that partially stabilized cross-effects of 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 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 can be reconstructed from this structure on the derivatives.
"Cross-effects and the classification of Taylor towers." Geom. Topol. 20 (3) 1445 - 1537, 2016. https://doi.org/10.2140/gt.2016.20.1445