On Triples, Operads, and Generalized Homogeneous Functors

Details On Triples, Operads, and Generalized Homogeneous Functors On Triples, Operads, and Generalized Homogeneous Functors

2004-01-25

arxiv.org/abs/math/0401346v1

Mathematics

Algebraic Topology

Scientific paper

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.

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