On operads, bimodules and analytic functors

Bimodules over Operads Characterize Morphisms

Let P be any operad. A P-bimodule R that is a P-cooperad induces a natural “fattening” of the category of P-(co)algebras, expanding the morphism sets while leaving the objects fixed. The morphisms in the resulting R-governed category of P-(co)algebras can be viewed as morphisms “up to R-homotopy” of P-(co)algebras. Let A denote the associative operad in the category of chain complexes. We defin...

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 mild...

Monoidal Functors, Acyclic Models and Chain Operads

We prove that for a topological operad P the operad of oriented cubical chains, C ∗ (P ), and the operad of singular chains, S∗(P ), are weakly equivalent. As a consequence, C ord ∗ (P ;Q) is formal if and only if S∗(P ;Q) is formal, thus linking together some formality results that are spread out in the literature. The proof is based on an acyclic models theorem for monoidal functors. We give ...

Benoit Fresse Modules over Operads and Functors

Right Modules over Operads and Functors

In the theory of operads we consider generalized symmetric power functors defined by sums of coinvariant modules. One observes classically that the symmetric functor construction provides an isomorphism from the category of symmetric modules to a split subcategory of the category of functors on dgmodules (if dg-modules form our ground category). The purpose of this article is to obtain a simila...