# Boardman-Vogt tensor product

Let $\mathbf{sSet}$ be the model category of simplicial sets and $\mathbf{Op}$ the model category of symmetric operads. Equipped with Boardman-Vogt tensor product $\otimes_{BV}$, the category $\mathbf{Op}$ is symmetric monoidal.

Questions:

1) is $(\mathbf{Op}, \otimes_{BV})$ a symmetric monoidal model category ?

2) let $P$ be a $E_{n}$-operad and $Q$ be a $E_{m}$-operad, is $P\otimes_{BV}Q$ a $E_{n+m}$-operad ?

Edit

1) $P$ and $Q$ are cofibrant symmetric operads.

2) a weak equivalence (fibration) of symmetic operads $f: A\rightarrow B$ is a level-wise weak equivalence (fibration) of simplicial sets $f_{n}: A(n)\rightarrow B(n)$.

3) the Boardman-Vogt tensor product $A\otimes_{BV} B$ is a (tricky) quotient of $A\sqcup B$ (the coproduct in $\mathbf{Op}$).

4) a $A\otimes_{BV}B$-algebra is a $A$-algebra in the category of $B$-algebras, or similarly a $B$-algebra in the category of $A$-algebras.

5) it is natural to ask if the category of $E_{n}$-algebras in the category of $E_{m}$-algebras is equivalent (in homotopical sense) to $E_{n+m}$-algebras. The question can be formulated as follows. It is true that
$$E_{n}\simeq E_{1}^{\otimes_{BV}^{n}}$$

• Does the notion of an $E_n$ operad include cofibrancy? Otherwise, $Ass\otimes_{BV} Ass = Comm$, I think. – Pavel Safronov Feb 22 '15 at 20:13
• yes, P and Q are cofibrant :) thanks for your remark. – Max Feb 22 '15 at 20:16