Boardman-Vogt tensor product



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 𝒫\mathcal{P} be an operad over a set of colors CC, and 𝒬\mathcal{Q} be an operad over a set of colors DD.

Their Boardman-Vogt tensor product P BVQP \otimes_{BV} Q is the operad whose set of colors is C×DC \times D, and whose operations are given by generators and relations as follows:

There is one generating operation for every pair (p,d)(p,d) with p𝒫(c 1,,c n;c)p \in \mathcal{P}(c_1, \cdots, c_n; c) and with dDd \in D, denoted

pd𝒫 BV𝒬((c 1,d),,(c n,d);(c,d)) p \otimes d \in \mathcal{P} \otimes_{BV} \mathcal{Q}( (c_1,d), \cdots, (c_n,d); (c,d) )

and for each pair (cq)(c \otimes q) with cCc \in C and q𝒬(d 1,,d n)q \in \mathcal{Q}(d_1, \cdots, d_n), denoted

(cq)𝒫 BV𝒬((c,d 1),,(c,d n);(c,d)) (c \otimes q) \in \mathcal{P}\otimes_{BV} \mathcal{Q}( (c, d_1), \cdots, (c, d_n); (c,d) )

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

  1. The tensor product c()c \otimes (-) with cCc \in C respects the composition in 𝒬\mathcal{Q}, and the tensor product ()d(-) \otimes d with dDd \in D respects the composition in 𝒫\mathcal{P} and both respect the action of the symmetric group on the operations.

    Equivalently this means that for all cCc \in C tensoring with cc extends to a morphism of operads

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

    and for all dDd \in D a morphism of operads

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


Closed monoidal structure


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

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

We write in the following

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

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


For P,QP, Q \in Operad, the internal hom operad [P,Q][P, Q] has

  • as colors the P-algebras in QQ;

  • as unary operations the PP-algebra homomorphisms in QQ.

See (Weiss, lemma 2.23).

We may therefore speak of [P,Q][P,Q] as being the operad of PP-algebras in QQ.


Write SetSet for the operad induced by the cartesian symmetric monoidal category structure on Set. Then for PP any operad, the vertices and unary operations of the internal hom operad [P,Set][P,Set] form the ordinary category of algebras over PP in SetSet.


For P 1,P 2,EOperadP_1, P_2, E \in Operad, the category of P 1P_1-algebras in P 2P_2-algebras in EE is equivalent to the category of P 2P_2-algebras in P 1P_1-algebras in EE.

In view of prop. 2 this is the statement of the closed symmetric monoidal structure (Operad, BC)(Operad, \otimes_{BC}):

[P 1,[P 2,E]][P 1 BVP 2,E][P 2 BVP 1,E][P 2,[P 1,E]]. [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]] \,.




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.

Revised on March 2, 2017 14:22:22 by Urs Schreiber (