ScalingStacks

A.8.7 Boardman-Vogt tensor product and Dunn additivity[00IW]

The \(\infty\)-category \(\mathrm{Op}\) of \(\infty\)-operads itself carries a symmetric monoidal structure, called the Boardman-Vogt tensor product uniquely characterized⁵⁷ by giving rise to an equivalence of \(\infty\)-operads for all \(\infty\)-operads \(\mathcal O, \mathcal O'\) and \(\mathcal P\): \[ \mathrm{Alg}_{\mathcal O}(\mathrm{Alg}_{\mathcal O'}(\mathcal P)) \simeq \mathrm{Alg}_{\mathcal O\otimes \mathcal O'}(\mathcal P).\] Equivalently, the Boardman-Vogt tensor product has \(\mathrm{Alg}_{-}(-)\) as its internal hom.

A fundamental theorem in the theory of \(\infty\)-operad is Dunn’s additivity theorem [Lur17, Thm. 5.1.2.2]: for \(n, m \geq 0\), there is an equivalence of \(\infty\)-operads \(\mathbb E_n \otimes \mathbb E_m \simeq \mathbb E_{n+m}\). In particular, using [00IX], an \(\mathbb E_{n+m}\)-algebra in an \(\infty\)-operad \(\mathcal O\) is equivalent to an \(\mathbb E_n\)-algebra in the \(\infty\)-operad of \(\mathbb E_{m}\)-algebras in \(\mathcal O\).