Functor Calculus and Operads March 13 – March 18 , 2011 MEALS

Speaker: Gregory Arone (Virginia) Title: Part 1: operads, modules and the chain rule Abstract: Let F be a homotopy functor between the categories of pointed topological spaces or spectra. By the work of Goodwillie, the derivatives of F form a symmetric sequence of spectra ∂∗F . This symmetric sequence determines the homogeneous layers in the Taylor tower of F , but not the extensions in the tower. In these two talks we will explore the following question: what natural structure does ∂∗F possess, beyond being a symmetric sequence? Our ultimate goal is to describe a structure that is sufficient to recover the Taylor tower of F from the derivatives. Such a description could be considered an extension of Goodwillie’s classification of homogeneous functors to a classification of Taylor towers. By a theorem of Ching, the derivatives of the identity functor form an operad. In the first talk we will see that the derivatives of a general functor form a bimodule (or a right/left module, depending on the source and target categories of the functor) over this operad. Koszul duality for operads plays an interesting role in the proof. As an application we will show that the module structure on derivatives is exactly what one needs to write down a chain rule for the calculus of functors. Let F be a homotopy functor between the categories of pointed topological spaces or spectra. By the work of Goodwillie, the derivatives of F form a symmetric sequence of spectra ∂∗F . This symmetric sequence determines the homogeneous layers in the Taylor tower of F , but not the extensions in the tower. In these two talks we will explore the following question: what natural structure does ∂∗F possess, beyond being a symmetric sequence? Our ultimate goal is to describe a structure that is sufficient to recover the Taylor tower of F from the derivatives. Such a description could be considered an extension of Goodwillie’s classification of homogeneous functors to a classification of Taylor towers. By a theorem of Ching, the derivatives of the identity functor form an operad. In the first talk we will see that the derivatives of a general functor form a bimodule (or a right/left module, depending on the source and target categories of the functor) over this operad. Koszul duality for operads plays an interesting role in the proof. As an application we will show that the module structure on derivatives is exactly what one needs to write down a chain rule for the calculus of functors. Title: Part 2: beyond the module structure Abstract: The (bi)module structure on derivatives is not sufficient to recover the Taylor tower of a functor. In the second talk we will refine the structure as follows. We will see that there is a naturally defined comonad on the category of (bi)-modules over the derivatives of the identity functor, and that the derivatives of a functor are a coalgebra over this comonad. From this coalgebra structure one can, in principle, reconstruct the Taylor tower of a functor. Thus this coalgebra structure seems to give a complete description of the structure possessed by the derivatives of a functor. We give an explicit description (as explicit as permitted by our current understanding) of the structure possessed by the derivatives of functors from Spectra to Spectra, from Spaces to Spectra and from Spaces to Spaces. An interesting example is the functor X !→ Σ∞Ω∞(E ∧X). Here E is a fixed spectrum, and the functor can be thought as a functor from either the category of Spaces or Spectra to the category of Spectra. The derivatives of this functor are given by the sequence E,E∧2, . . . , E∧n, . . .. The fact that this sequence is the sequence of derivatives of a functor seems to tell us something about the structure possessed by spectra in general. In particular it tells us that spectra possess a natural structure of a restricted algebra over the Lie operad (an observation first made by Bill Dwyer). Speaker: Mark Behrens (MIT) Title: Survey of the Goodwillie tower of the identity I Abstract: The Goodwillie tower of the identity functor from spaces to spaces is a powerful tool for understanding unstable homotopy from the stable point of view. I will describe the derivatives of this functor, which were studied by Johnson and Arone-Mahowald. I will also explain the Arone-Mahowald computation of the homology of the layers of the tower evaluated on spheres. These layers were shown by Arone-Dwyer to be equivalent to stunted versions of the L(k) spectra studied by Kuhn, Mitchell, Priddy, and others. Associated to the Goodwillie tower is a spectral sequence which computes the unstable homotopy groups of a space from the stable homotopy groups of the layers. I will explain consequences for unstable vk-periodic homotopy discovered by Arone and Mahowald. I will also discuss relations with the EHP sequence at the prime 2. Specifically, differentials in the Goodwillie spectral sequence can often be computed in terms of Hopf invariants, and differentials in the EHP sequence can often be computed in terms of attaching maps in the L(k)-spectra. Title: Survey of the Goodwillie tower of the identity II Abstract: I will recall Ching’s operad structure on the derivatives of the identity. Coupled with operadic structures recently discovered by Arone-Ching, this gives an action of a certain algebra of Dyer-Lashof-like