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Cross-effects and the classification of Taylor towers

Gregory Arone and Michael Ching

Geometry & Topology 20 (2016) 1445–1537

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.

Goodwillie calculus, homotopy functors, operads
Mathematical Subject Classification 2010
Primary: 55P65, 55P48
Secondary: 18D50
Received: 14 April 2014
Revised: 30 March 2015
Accepted: 18 July 2015
Published: 4 July 2016
Proposed: Mark Behrens
Seconded: Haynes Miller, Bill Dwyer
Gregory Arone
Department of Mathematics
Stockholm University
SE-106 91 Stockholm
Michael Ching
Department of Mathematics and Statistics
Amherst College
Amherst, MA 01002
United States