nLab Boardman-Vogt tensor product

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

In the 1-category of symmetric operads

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 (…).

In the $\infty$-category of $\infty$-operads

Let $\mathrm{Cat}_{\infty,/\mathbb{F}_*}^{\mathrm{Int}-\mathrm{cocart}} \subset \mathrm{Cat}_{\infty,/\mathbb{F}_*}$ denote the (non-full) subcategory of functors to finite pointed sets which posses cocartesian lifts over inert morphisms.

The full subcategory $\mathrm{Op}_\infty \subset \mathrm{Cat}_{\infty,/\mathbb{F}_*}^{\mathrm{Int}-\mathrm{cocart}}$ spanned by the $\infty$-operads of Lurie is localizing; write $L_{\mathrm{Op}}: \mathrm{Cat}_{\infty,/\mathbb{F}_*}^{\mathrm{Int}-\mathrm{cocart}} \rightarrow \mathrm{Op}_\infty$ for its localization functor.

Then, given $\mathcal{O}^{\otimes},\mathcal{P}^{\otimes} \in \mathrm{Op}_\infty$, their Boardman-Vogt tensor product is the localization

$$\mathcal{O}^{\otimes} \otimes_{\mathrm{BV}} \mathcal{P}^{\otimes} := L_{\mathrm{Op}} \left( \mathcal{O}^{\otimes} \times \mathcal{P}^{\otimes} \rightarrow \mathbb{F}_* \times \mathbb{F}_* \xrightarrow{\wedge} \mathbb{F}_* \right).$$

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

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

The $\infty$-categorical universal property of the BV-tensor product

An arrow $\mathcal{O}^{\otimes} \times \mathcal{P}^{\otimes} \rightarrow \mathcal{Q}^{\otimes}$ is a bifunctor of $\infty$-operads if it extends to a commutative diagram

sending pairs of (cocartesian lifts of) inert morphisms in $\mathcal{O}^{\otimes} \times \mathcal{P}^{\otimes}$ to (cocartesian lifts of) inert morphisms in $\mathcal{Q}$.

Let $\mathrm{BiFun}(\mathcal{O}^{\otimes},\mathcal{P}^{\otimes}; \mathcal{Q}^{\otimes}) \subset \mathrm{Fun}_{/\mathbb{F}_*}(\mathcal{O}^{\otimes} \times \mathcal{P}^{\otimes}, \mathcal{Q}^{\otimes})$ be the full subcategory spanned by functors satisfying the above inert morphism condition.

Proposition

There is an equivalence of $\infty$-categories

$$\mathrm{Alg}_{\mathcal{O}^{\otimes} \otimes_{\mathrm{BV}} \mathcal{P}^{\otimes}}(\mathcal{Q}^{\otimes}) \simeq \mathrm{BiFun}(\mathcal{O}^{\otimes},\mathcal{P}^{\otimes}; \mathcal{Q}^{\otimes}) .$$

The symmetric monoidal $\infty$-categorical universal property

The following is Theorem E of Barkan-Steinebrunner 23

Theorem

The canonical tensor product of symmetric monoidal $\infty$-categories uniquely restricts to a tensor product on $\mathrm{Op}_\infty$ such that the symmetric monoidal envelope is a symmetric monoidal functor.

This is not the symmetric monoidal structure constructed in Higher algebra; however, both it and that of Higher algebra have tensor functors satisfying the above universal property, so their tensor functors agree.

Lurie’s tensor product comes from an ad-hoc construction involving a left-derived functor from a seldom-used monoidal model category, which doesn’t obviously satisfy many nice properties; thus it is likely that the above theorem constructs the “correct” coherences on $\otimes_{\BV}$, and those in Higher Algebra may be “incorrect.”

Connectivity and the Eckmann-Hilton argument

Let $\mathcal{A}_2^{\otimes}$ be the free unital (∞,1)-operad with a binary operation (so that it’s Stasheff’s 2nd operad, i.e. $\mathcal{A}_2$-algebras are unital magmas). The Eckmann-Hilton argument states that the canonical map of (∞,1)-operads $\mathcal{A}_2^{\otimes} \otimes_{\mathrm{BV}} \mathcal{A}_2^{\otimes} \rightarrow \mathrm{Comm}^{\otimes}$ induces an equivalence

$$\mathrm{Alg}_{\mathcal{A}_2} \mathrm{Alg}_{\mathcal{A}_2}^{\otimes}(\mathcal{C}) \simeq \mathrm{Alg}_{\mathcal{A}_2 \otimes_{\mathrm{BV}} \mathcal{A}_2}(\mathcal{C}) \rightarrow \mathrm{CAlg}(\mathcal{C})$$

whenever $\mathcal{C}$ is a symmetric monoidal 1-category.

We say that a reduced (∞,1)-operad $\mathcal{O}^{\otimes}$ is $n$-connected if the canonical map $\mathcal{O}^{\otimes} \rightarrow \mathrm{Comm}^{\otimes}$ induces equivalences on the respective categories of algebras in symmetric monoidal $n$-categories.

The following result is proved in Schlank-Yanovski 19, where it is called the $\infty$-categorical Eckmann-Hilton argument.

Theorem

If $\mathcal{O}^{\otimes}$ is $d_{\mathcal{O}}$-connected and $\mathcal{P}^{\otimes}$ is $d_{\mathcal{P}}$-connected, then the Boardman-Vogt tensor product $\mathcal{O}^{\otimes} \otimes_{\mathrm{BV}} \mathcal{P}^{\otimes}$ is $(d_{\mathcal{O}} + d_{\mathcal{P}} + 2)$-connected.

Examples

Infinite tensor products of reduced $\infty$-operads are $\mathrm{Comm}$.

Let $\mathcal{O}^{\otimes}$ be a reduced $\infty$-operad other than $\mathbb{E}_0$, so that there is a canonical map of operads $\mathrm{triv}^{\otimes} \rightarrow \mathcal{O}^{\otimes}$, inducing a teloscope

$$\mathcal{O}^{\otimes \infty} \coloneqq \colim \left( \mathcal{O}^{\otimes} \rightarrow \mathcal{O}^{\otimes 2} \rightarrow \mathcal{O}^{\otimes 3} \rightarrow \cdots \right)$$

The above connectivity bound implies the following corollary, which may be viewed as an operadic Eilenberg swindle.

Corollary

The canonical map $\mathcal{O}^{\otimes \infty} \rightarrow \mathrm{Comm}^{\otimes}$ is an equivalence.

Dunn’s additivity theorem

Let $C_n$ be the topological little n-cubes operad. Then, the canonical maps $C_n,C_m \rightarrow C_{n+m}$ together yield a map $\mu:C_n \otimes C_m \rightarrow C_{n+m}$. In the case $n = 1$, Dunn 88 constructs a diagram

where $D_{n+m}$ us the operad of decomposable little $(n+m)$-cubes and $C_{n} | C_{m}$ is the image of $\mu$. The difficult statement in Dunn 88 is the statement that $\mu'$ is a local $\Sigma$-equivalence. This implies the following.

Theorem

The topological operads $C_n \otimes C_m \simeq C_{n+m}$ are related by a zigzag of local $\Sigma$-equivalences.

This theorem is slightly unusual, as the operad $C_n$ is not cofibrant, and the tensor product in the statement is underived. Simpler proofs of this statement involve the thesis Brinkmeier 00 and the recent paper Barata-Moerdijk 22.

The corresponding statement for the derived tensor product is proved as Theorem 5.1.2.2 in Higher Algebra.

Theorem

The canonical map of $\infty$-operads $\mathbb{E}_{n}^{\otimes} \otimes \mathbb{E}_{m}^{\otimes} \rightarrow \mathbb{E}_{n + m}^{\otimes}$ is an equivalence; hence for any symmetric monoidal $\infty$-category $\mathcal{C}$, the pullback functor

$$\mathrm{Alg}^{\otimes}_{\mathbb{E}_{n+m}}(\mathcal{C}) \rightarrow \mathrm{Alg}_{\mathbb{E}_n}^{\otimes} \mathrm{Alg}_{\mathbb{E}_m}^{\otimes}(\mathcal{C})$$

is an equivalence.

Related concepts

• The Boardman-Vogt tensor product extends from operads to a tensor product on dendroidal sets. See there for more details.

• tensor product of monads

References

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

• Ittay Weiss, From Operads to Dendroidal Sets, in Mathematical Foundations of Quantum Field and Perturbative String Theory AMS (2011) (arXiv:1012.4315)

see around def. 2.21 there.

Underived proofs of Dunn’s additivity theorem include

• Gerald Dunn, Tensor product of operads and iterated loop spaces. (1988) (pdf)

• Michael Brinkmeier, On Operads. PhD thesis, Universitat Osnabrüeck, 2000.

• Miguel Barata?, Ieke Moerdijk, On the additivity of the little cubes operads (2022) (arXiv:2205.12875)

The $\infty$-categorical universal property is Definition 2.2.5.3 in the following textbook; an $\infty$-operadic version of Dunn’s additivity theorem is Theorem 5.1.2.2.

• Higher Algebra.

It is reviewed in the form used above in the lecture notes

• Rune Haugseng, An allegedly somewhat friendly introduction to $\infty$-operads (pdf)

It was realized to be compatible with the symmetric monoidal envelope in

• Shaul Barkan, Jan Steinebrunner, Segalification and the Boardman-Vogt tensor product, arXiv:2301.08650.

The ∞-categorical Eckmann-Hilton argument is due to

• Tomer Schlank, Lior Yanovski, The ∞-Categorical Eckmann-Hilton Argument, Algebr. Geom. Topol. 19 (2019) 3119-3170 (arXiv:1808.06006, doi:10.2140/agt.2019.19.3119)