# nLab Boardman-Vogt tensor product

Contents

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

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.

## Definition

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

$p \otimes d \in \mathcal{P} \otimes_{BV} \mathcal{Q}( (c_1,d), \cdots, (c_n,d); (c,d) )$

and for each pair $(c,q)$ with $c \in C$ and $q \in \mathcal{Q}(d_1, \cdots, d_n; d)$, denoted

$c \otimes q \in \mathcal{P}\otimes_{BV} \mathcal{Q}( (c, d_1), \cdots, (c, d_n); (c,d) )$

for all $n \in \mathbb{N}$. These are subject to the following relations

1. 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

$\mathcal{Q} \to \{c\} \otimes_{BV}\mathcal{Q}$

and for all $d \in D$ a morphism of operads

$\mathcal{P} \to \mathcal{P} \otimes_{BV} \{d\} \,.$
2. The operations in $\mathcal{P}$ and $\mathcal{Q}$ distribute over each other in $\mathcal{P} \otimes_{BV} \mathcal{Q}$ in the evident sense (…).

## Properties

### Closed monoidal structure

###### Proposition

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

###### Corollary
• The BV tensor products preserves colimits of operads in each variable separately.

We write in the following

$[-,-] : Operad^{op} \times Operad \to Operad$

for the corresponding internal hom (leaving a subscrip “${}_{BV}$” implicit.)

###### Proposition

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 $Q$.

###### Example

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 $Set$.

###### Corollary

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 $(Operad, \otimes_{BC})$:

$[P_1, [P_2, E]] \simeq [P_1 \otimes_{BV} P_2, E] \simeq [P_2 \otimes_{BV} P_1, E] \simeq [P_2, [P_1, E]] \,.$

## Examples

(…)

The original reference is

• Michael Boardman, Rainer Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, Vol. 347, Springer-Verlag, (1973).

A review is in

see around def. 2.21 there.