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.
Categories: math.AT