symmetric monoidal (∞,1)-category of spectra
The Boardman-Vogt tensor product is a natural tensor product on symmetric operads. It makes the category Operad of colored symmetric operads over Set into a closed monoidal category.
All operads in the following are colored symmetric operads enriched over Set, equivalently symmetric multicategories.
Let $\mathcal{P}$ be an operad over a set of colors $C$, and $\mathcal{Q}$ be an operad over a set of colors $D$.
Their Boardman-Vogt tensor product $\mathcal{P} \otimes_{BV} \mathcal{Q}$ is the operad whose set of colors is $C \times D$, and whose operations are given by generators and relations as follows:
There is one generating operation for every pair $(p,d)$ with $p \in \mathcal{P}(c_1, \cdots, c_n; c)$ and with $d \in D$, denoted
and for each pair $(c,q)$ with $c \in C$ and $q \in \mathcal{Q}(d_1, \cdots, d_n; d)$, denoted
for all $n \in \mathbb{N}$. These are subject to the following relations
The tensor product $c \otimes (-)$ with $c \in C$ respects the composition in $\mathcal{Q}$, and the tensor product $(-) \otimes d$ with $d \in D$ respects the composition in $\mathcal{P}$ and both respect the action of the symmetric group on the operations.
Equivalently this means that for all $c \in C$ tensoring with $c$ extends to a morphism of operads
and for all $d \in D$ a morphism of operads
The operations in $\mathcal{P}$ and $\mathcal{Q}$ distribute over each other in $\mathcal{P} \otimes_{BV} \mathcal{Q}$ in the evident sense (…).
Equipped with the Boardman-Vogt tensor product, Operad is a closed symmetric monoidal category.
See for instance the proof provided in (Weiss, theorem 2.22).
This implies directly several useful statements about the BV-tensor product
We write in the following
for the corresponding internal hom (leaving a subscrip “${}_{BV}$” implicit.)
For $P, Q \in$ Operad, the internal hom operad $[P, Q]$ has
as colors the P-algebras in $Q$;
as unary operations the $P$-algebra homomorphisms in $Q$.
See (Weiss, lemma 2.23).
We may therefore speak of $[P,Q]$ as being the operad of $P$-algebras in $Q$.
Write $Set$ for the operad induced by the cartesian symmetric monoidal category structure on Set. Then for $P$ any operad, the vertices and unary operations of the internal hom operad $[P,Set]$ form the ordinary category of algebras over $P$ in $Set$.
For $P_1, P_2, E \in Operad$, the category of $P_1$-algebras in $P_2$-algebras in $E$ is equivalent to the category of $P_2$-algebras in $P_1$-algebras in $E$.
In view of prop. this is the statement of the closed symmetric monoidal structure $(Operad, \otimes_{BC})$:
(…)
The Boardman-Vogt tensor product extends from operads to a tensor product on dendroidal sets. See there for more details.
The original reference is
A review is in
see around def. 2.21 there.
Last revised on July 2, 2020 at 21:22:48. See the history of this page for a list of all contributions to it.