The Boardman-Vogt tensor product of operadic bimodules

We define and study a lift of the Boardman-Vogt tensor product of operads to bimodules over operads. Introduction Let Op denote the category of symmetric operads in the monoidal category S of simplicial sets. The Boardman-Vogt tensor product [3] −⊗− : Op× Op→ Op, which endows the category Op with a symmetric monoidal structure, codifies interchanging algebraic structures. For all P,Q ∈ Op, a (P ⊗ Q)-algebra can be viewed as a P-algebra in the category of Q-algebras or as a Q-algebra in the category of P-algebras. In this article we lift the Boardman-Vogt tensor product to the category of composition bimodules over operads and study the properties of the lifted tensor product. The lifted Boardman-Vogt tensor product is an essential tool in two articles that we are currently preparing. One of these articles concerns the space of configurations in a product of framed manifolds, while in the second we generalize [7], building an “operadic” model for the space of long links in R for m ≥ 4. Let P,Q ∈ Op. Let BimodP,Q denote the category of composition bimodules over P on the left and Q on the right. An object of BimodP,Q is a symmetric sequence in S endowed with a left action of P and a right action of Q, with respect to the composition monoidal product ◦ of symmetric sequences, which are appropriately compatible. A pair of operad morphisms φ : P→ P′ and ψ : Q→ Q′ gives rise to a functor (φ∗, ψ∗) : BimodP′,Q′ → BimodP,Q by restriction of coefficients. Gathering together all composition bimodules over all operads, we form a category Bimod. An object of Bimod is a composition bimodule over a pair of operads (P,Q). A morphism in Bimod from a (P,Q)-bimodule M to a (P′,Q′)-bimodule M′ consists of a triple (φ,ψ, f), where φ : P → P′ and ψ : Q → Q′ are operad morphisms, and f : M → (φ∗, ψ∗)(M′) is a morphism of (P,Q)-bimodules. There is an obvious projection functor Π : Bimod→ Op× Op, which admits a section Γ : Op× Op→ Bimod : (P,Q) 7→ P ◦ Q, 2010 Mathematics Subject Classification. Primary: 18D50; Secondary: 18D10, 55P48.