Michael Ching

Bar constructions for topological operads and the Goodwillie derivatives of the identity

We describe a cooperad structure on the simplicial bar construction on a reduced operad of based spaces or spectra and, dually, an operad structure on the cobar construction on a cooperad. We also show that if the homology of the original operad (respectively, cooperad) is Koszul, then the homology of the bar (respectively, cobar) construction is the Koszul dual. We use our results to construct an operad structure on the partition poset models for the Goodwillie derivatives of the identity functor on based spaces and show that this induces the ‘Lie’ operad structure on the homology groups of these derivatives. We also extend the bar construction to modules over operads (and, dually, to comodules over cooperads) and show that a based space naturally gives rise to a left module over the operad formed by the derivatives of the identity.

Bar–cobar duality for operads in stable homotopy theory

We extend bar–cobar duality, defined for operads of chain complexes by Getzler and Jones, to operads of spectra in the sense of stable homotopy theory. Our main result is the existence of a Quillen equivalence between the category of reduced operads of spectra (with the projective model structure) and a new model for the homotopy theory of cooperads of spectra. The crucial construction is of a weak equivalence of operads between the Boardman–Vogt W-construction for an operad P, and the cobar–bar construction on P. This weak equivalence generalizes a theorem of Berger and Moerdijk that says the W- and cobar–bar constructions are isomorphic for operads of chain complexes.Our model for the homotopy theory of cooperads is based on ‘pre-cooperads’. These can be viewed as cooperads in which the structure maps are zigzags of maps of spectra that satisfy coherence conditions. Our model structure on pre-cooperads is such that every object is weakly equivalent to an actual cooperad, and weak equivalences between cooperads are detected in the underlying symmetric sequences.We also interpret our results in terms of a ‘derived Koszul dual’ for operads of spectra, which is analogous to the Ginzburg–Kapranov dg-dual. We show that the double derived Koszul dual of an operad P is equivalent to P (under some finiteness hypotheses) and that the derived Koszul construction preserves homotopy colimits, finite homotopy limits and derived mapping spaces for operads.

A CHAIN RULE FOR GOODWILLIE DERIVATIVES OF FUNCTORS FROM SPECTRA TO SPECTRA

We prove a chain rule for the Goodwillie calculus of functors from spectra to spectra. We show that the (higher) derivatives of a composite functor F G at a base object X are given by taking the composition product (in the sense of symmetric sequences) of the derivatives of F at G(X) with the derivatives of G at X. We also consider the question of finding Pn(F G), and give an explicit formula for this when F is homogeneous.

Cross-effects and the classification of Taylor towers

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 spectr ...