On the Goodwillie Derivatives of the Identity in Structured Ring Spectra

Abstract: Functor calculus was introduced by Goodwillie as a means for analyzing homotopy functors between suitable model categories. One noteworthy facet is that "nice" functors $F\colon \mathsf{C}\to \mathsf{D}$ are determined by a certain symmetric sequence called the derivatives of $F$.

This sequence of derivatives is known to posses much structure: for instance, the derivatives of the identity functor on the category of based topological spaces is an operad, as first shown by Ching. It is further expected that a result of this type should hold in any suitable model category, and in particular conjectured that the derivatives of the identity on the category of algebras over an operad $\mathcal{O}$ in spectra should be equivalent to $\mathcal{O}$ as operads.

In this talk we produce an intrinsic "homotopy-coherent" operad structure for the derivatives of the identity which is equivalent to that on $\mathcal{O}$, thus resolving the above conjecture. Along the way we will discuss the necessary background of functor calculus and algebras over operads of spectra. Our method is to induce a homotopy coherent operadic pairing on the derivatives by a suitable pairing on the cosimplicial resolution offered by the stabilization adjunction for $\mathcal{O}$-algebras. Time permitting, we will provide some other applications of our techniques such as a highly homotopy-coherent chain rule for functors of structured ring spectra.

algebraic topologydifferential geometrygeneral topologygeometric topology

Audience: researchers in the topic

GOATS

Series comments: Description: A series of online mini-conferences for graduate students in Geo

Conference will be on Zoom and simultaneously livestreamed to Youtube. Zoom link is obtained by registering on the website, the livestream link will be posted closer to conference date.