On Triples, Operads, and Generalized Homogeneous Functors

We study the splitting of the Goodwillie towers of functors in various settings. In particular, we produce splitting criteria for functors F : A → MA from a pointed category with coproducts to A-modules in terms of differentials of F . Here A is a commutative S-algebra. We specialize to the case when A is the category of a-algebras for an operad a and F is the forgetful functor, and derive milder splitting conditions in terms of the derivative of F . In addition, we describe how triples induce operads, and prove that, roughly speaking, a triple T is naturally equivalent to the product of its Goodwillie layers if and only if it is an algebra over its induced operad.