symmetric monoidal (∞,1)-category of spectra
Let be an operad over a set of colors , and be an operad over a set of colors .
Their Boardman-Vogt tensor product is the operad whose set of colors is , and whose operations are given by generators and relations as follows:
There is one generating operation for every pair with and with , denoted
and for each pair with and , denoted
for all . These are subject to the following relations
The tensor product with respects the composition in , and the tensor product with respects the composition in and both respect the action of the symmetric group on the operations.
Equivalently this means that for all tensoring with extends to a morphism of operads
and for all a morphism of operads
The operations in and distribute over each other in in the evident sense (…).
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 “” implicit.)
For Operad, the internal hom operad has
as colors the P-algebras in ;
as unary operations the -algebra homomorphisms in .
See (Weiss, lemma 2.23).
We may therefore speak of as being the operad of -algebras in .
Write for the operad induced by the cartesian symmetric monoidal category structure on Set. Then for any operad, the vertices and unary operations of the internal hom operad form the ordinary category of algebras over in .
For , the category of -algebras in -algebras in is equivalent to the category of -algebras in -algebras in .
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.