2020/04/06 by Clark, Duncan A.
#Algebraic Topology (math.AT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2004.02812
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 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.