Infinity‐operads and Day convolution in Goodwillie calculus

We prove two theorems about Goodwillie calculus and use those theorems to describe new models for Goodwillie derivatives of functors between pointed compactly generated ∞‐categories. The first theorem says that the construction of higher derivatives for spectrum‐valued functors is a Day convolution of copies of the first derivative construction. The second theorem says that the derivatives of any functor can be realized as natural transformation objects for derivatives of spectrum‐valued functors. Together these results allow us to construct an ∞‐operad that models the derivatives of the identity functor on any pointed compactly generated ∞‐category. Our main example is the ∞‐category of algebras over a stable ∞‐operad, in which case we show that the derivatives of the identity essentially recover the same ∞‐operad, making precise a well‐known slogan in Goodwillie calculus. We also describe a bimodule structure on the derivatives of an arbitrary functor, over the ∞‐operads given by the derivatives of the identity on the source and target, and we conjecture a chain rule that generalizes previous work of Arone and the author in the case of functors of pointed spaces and spectra.