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 $F: \A \to M_A$fromapointedcategorywithcoproductsto$A$-modules in terms of differentials of $F$. Here $A$ is a commutative $S$-algebra. We specialize to the case when $\text{A}$ is the category of $\text{a}$-algebras for an operad $\text{a}$ 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.