An Intrinsic Operad Structure for the Derivatives of the Identity

A long standing slogan in Goodwillie's functor calculus is that the derivatives of the identity functor on a suitable model category should come equipped with a natural operad structure. A result of this type was first shown by Ching for the category of based topological spaces. It has long been expected that in the category of algebras over a reduced operad $\mathcal{O}$ of spectra that the derivatives of the identity should be equivalent to $\mathcal{O}$ as operads.
 
In this talk I will discuss my recent work which gives a positive answer to the above conjecture. My method is to induce a ``highly homotopy coherent'' operad structure on the derivatives of the identity via an pairing of underlying cosimplicial objects with respect to the box product. This cosimplicial object naturally arises by analyzing the derivatives of the Bousfield-Kan cosimplicial resolution of the identity via the stabilization adjunction for $\mathcal{O}$-algebras. Time permitting, I will describe some additional applications of these box product pairings. In particular, I will show how a similar box product pairing may be utilized to provide a new description of an operad structure on the derivatives of the identity in spaces.