On Triples, Operads, and Generalized Homogeneous Functors

McCarthy, Randy; Minasian, Vahagn

On Triples, Operads, and Generalized Homogeneous Functors

Authors

Description

We study the splitting of the Goodwillie towers of functors in various settings. In particular, we produce splitting criteria for functors $F: \A \to M_A$ 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.