On Triples, Operads, and Generalized Homogeneous Functors

TL;DR

This paper investigates the splitting of Goodwillie towers for functors in algebraic and categorical settings, providing criteria based on differentials and derivatives, and explores the relationship between triples and operads.

Contribution

It introduces new splitting criteria for functors from pointed categories to modules, especially for algebraic structures governed by operads, and links triples to operad algebras.

Findings

Splitting criteria depend on differentials and derivatives of functors.

Triples are shown to induce operads, with equivalence to their Goodwillie layers.

Milder splitting conditions are derived for forgetful functors from algebra categories.

Abstract

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.