On the Goodwillie derivatives of the identity in structured ring spectra

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.

arXiv commentFinal version. To appear in Tbilisi Math. Journal: Special Issue on Homotopy Theory, Spectra, and Structured Ring Spectra

References in corpus (11)

• Rings, modules, and algebras in infinite loop space theory
• Homology and Cohomology of E-infinity Ring Spectra
• A general framework for homotopic descent and codescent
• Symmetric homotopy theory for operads
• Operad bimodules, and composition products on Andre-Quillen filtrations of algebras
• On Triples, Operads, and Generalized Homogeneous Functors
• Suspension spectra and higher stabilization
• Iterated delooping and desuspension of structured ring spectra
• Infinity-operads and Day convolution in Goodwillie calculus
• A Monoidal Model for Multilinearization
• Fibration theorems for TQ-completion of structured ring spectra

Cited by in corpus (3)

• Infinity-operads and Day convolution in Goodwillie calculus
• TQ-completion and the Taylor tower of the identity functor
• The partition poset complex and the Goodwillie derivatives of the identity in spaces