Duoidal $\infty$-categories of operadic modules

Abstract

In this paper we study duoidal structures on $\infty$-categories of operadic modules. Let ${\mathcal{O}}^{\otimes }$ be a small coherent $\infty$-operad and let ${\mathcal{P}}^{\otimes }$ be an $\infty$-operad. If a $\mathcal{P}\otimes \mathcal{O}$-monoidal $\infty$-category ${\mathcal{C}}^{\otimes }$ has a sufficient supply of colimits, then we show that the $\infty$-category ${\mathrm{Mod}}_A^{\mathcal{O}}(\mathcal{C})$ of $\mathcal{O}$-$A$-modules in ${\mathcal{C}}^{\otimes }$ has a structure of $(\mathcal{P},\mathcal{O})$-duoidal $\infty$-category for any $\mathcal{P}\otimes \mathcal{O}$-algebra object $A$.

21 pages