Michael Ching's Home Page
I did my PhD at MIT, working with Haynes Miller. My thesis was on Bar constructions for topological operads and the Goodwillie derivatives of the identity.
This project is to extend Kuhn's results on the splitting of Taylor towers of functors of K(n)-local spectra to the unstable case. We do not expect split Taylor towers in this case, but we do expect the Taylor towers to be determined by the Goodwillie derivatives together with the bimodule structure we constructed on those derivatives. The idea is to apply Kuhn's theorem on vanishing Tate cohomology to the "fake" Taylor towers arising from certain Ext-objects in the category of bimodules over the derivatives of the identity functor.
This is intended to explore fully the Koszul duality used to construct module structures on Goodwillie derivatives. If P is an operad in spectra, we have a corresponding bar-cooperad BP. The idea is that the homotopy category P-modules (either right, left, or bi-) is equivalent to that of BP-comodules. Taking a Spanier-Whitehead dual, this should be equivalent to a category of DBP-modules. This is strictly true only on when restricted to termwise-finite objects. For a more general result, we use pro-spectra.
This is an attempt to apply calculate the Taylor tower and Goodwillie derivatives of algebraic K-theory viewed as a functor from associative (or commutative) S-algebras to spectra. This fits into the framework developed in "Operads and chain rules for the calculus of functors" (below) and makes use of results of Basterra-Mandell on the stabilization of categories of S-algebras, and of Lindenstrauss-McCarthy on Taylor towers of other algebraic K-theory functors.
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.
We consider the composition product of symmetric sequences in the case where the underlying symmetric monoidal structure does not commute with coproducts. Even though this composition product is not a monoidal structure on symmetric sequences, it has enough properties to be able to define monoids (which are then operads on the underlying category) and make a bar construction. The main benefit of this work is in the dual setting, where it allows us to define a cosimplicial cobar construction for cooperads.
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 FG 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(FG), and give an explicit formula for this when F is homogeneous.
We study the structure possessed by the Goodwillie derivatives of a pointed homotopy functor of based topological spaces. These derivatives naturally form a bimodule over the operad consisting of the derivatives of the identity functor. We then use these bimodule structures to give a chain rule for higher derivatives in the calculus of functors, extending that of Klein and Rognes. This chain rule expresses the derivatives of FG as a derived composition product of the derivatives of F and G over the derivatives of the identity. There are two main ingredients in our proofs. Firstly, we construct new models for the Goodwillie derivatives of functors of spectra. These models allow for natural composition maps that yield operad and module structures. Then, we use a cosimplicial cobar construction to transfer this structure to functors of topological spaces. A form of Koszul duality for operads of spectra plays a key role in this.
This is a summary of a talk given at the Conference on Pure and Applied Topology on the Isle of Skye in June 2005. We describe a relationship between right modules over the operad formed by the Goodwillie derivatives of the identity functor, and configuration spaces on manifolds. The missing step in turning this project into a paper is a proper understanding of the self-Koszul-duality of the Fulton-MacPherson operads of compactified configuration spaces.
This paper describes models for the cross-effects of homotopy functors given in terms of spaces of trees. These link closely to the methods of the paper "Bar constructions for topological operads and the Goodwillie derivatives of the identity", but I do not have any useful applications for these models.
This was my first attempt as a graduate student to understand what operations exist on the homotopy groups of a simplicial algebra over an operad, in a sense generalizing the Cartan-Bousfield-Dwyer operations on simplicial commutative algebras. It contains some partial results, but not a full description.