[Paper Review] On the Goodwillie derivatives of the identity in structured ring spectra

Duncan A. Clark|arXiv (Cornell University)|Apr 6, 2020

Homotopy and Cohomology in Algebraic Topology44 references5 citations

TL;DR

This paper constructs a highly homotopy coherent operad structure on the Goodwillie derivatives of the identity functor in the category of algebras over a reduced operad 𝒪 in spectra. It proves that these derivatives are weakly equivalent to 𝒪 itself as highly homotopy coherent operads, resolving a conjecture by Arone and Ching. The construction uses Bousfield-Kan cosimplicial resolutions and strong cartesianity estimates to relate the derivatives to totalizations of cosimplicial diagrams.

ABSTRACT

The aim of this paper is three-fold: (i) we construct a naturally occurring highly homotopy coherent operad structure on the derivatives of the identity functor on structured ring spectra which can be described as algebras over an operad $\mathcal{O}$ in spectra, (ii) we prove that every connected $\mathcal{O}$-algebra has a naturally occurring left action of the derivatives of the identity, and (iii) we show that there is a naturally occurring weak equivalence of highly homotopy coherent operads between the derivatives of the identity on $\mathcal{O}$-algebras and the operad $\mathcal{O}$. Along the way, we introduce the notion of $\mathbf{N}$-colored operads with levels which -- by construction -- provides a precise algebraic framework for working with and comparing highly homotopy coherent operads, operads, and their algebras.

Motivation & Objective

• To construct a natural, highly homotopy coherent operad structure on the Goodwillie derivatives of the identity functor in the category of 𝒪-algebras.
• To show that every connected 𝒪-algebra admits a natural left action by these derivatives.
• To establish a weak equivalence between the derivatives of the identity and the original operad 𝒪 as highly homotopy coherent operads.
• To introduce N-colored operads with levels as a framework for comparing homotopy coherent operads and their algebras.

Proposed method

• Replacing the identity functor with the Bousfield-Kan cosimplicial resolution via the stabilization adjunction (Q, U) for 𝒪-algebras.
• Using strong cartesianity estimates from Blomquist and Ching-Harper to express the derivatives as a homotopy limit of a cosimplicial diagram.
• Constructing a cosimplicial diagram of symmetric sequences whose totalization models the derivatives.
• Defining a highly homotopy coherent operad structure via the totalization of a cosimplicial operad built from 𝒪.
• Proving that the totalization of the cosimplicial diagram is weakly equivalent to 𝒪 via a zigzag of weak equivalences.
• Introducing N-colored operads with levels to formalize comparisons between homotopy coherent operads, operads, and their algebras.

Experimental results

Research questions

• RQ1 Can the Goodwillie derivatives of the identity functor on 𝒪-algebras be endowed with a natural, highly homotopy coherent operad structure?
• RQ2 Is there a weak equivalence between the derivatives of the identity and the original operad 𝒪 as highly homotopy coherent operads?
• RQ3 How can one define a left action of the derivatives of the identity on connected 𝒪-algebras?
• RQ4 What algebraic framework allows precise comparison of highly homotopy coherent operads and their algebras?
• RQ5 Can the TQ-completion of a 0-connected 𝒪-algebra be described as an algebra over the derivatives of the identity?

Key findings

• The derivatives of the identity functor on 𝒪-algebras admit a natural, highly homotopy coherent operad structure.
• Every connected 𝒪-algebra naturally carries a left action by the derivatives of the identity.
• The derivatives of the identity are weakly equivalent to the original operad 𝒪 as highly homotopy coherent operads.
• The equivalence is established via the totalization of a cosimplicial diagram built from the operad 𝒪 and its resolutions.
• The TQ-completion of any 0-connected 𝒪-algebra is equivalent to an algebra over the derivatives of the identity.
• The framework of N-colored operads with levels provides a precise algebraic setting for comparing homotopy coherent structures.