On the Goodwillie derivatives of the identity in structured ring spectra

D. A. Clark

The aim of this paper is three-fold: (i) we construct a natural 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 O in spectra, (ii) we prove that every connected 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 O-algebras and the operad O.
Along the way, we introduce the notion of 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. 

Tbilisi Mathematical Journal, Special Issue (HomotopyTheorySpectra - 2020), pp. 119-166