Duoidal $\infty$-categories of operadic modules

TL;DR

This paper explores duoidal structures on $ abla$-categories of operadic modules, establishing conditions under which these categories acquire a duoidal $ abla$-category structure in the context of $ abla$-operads and $ abla$-algebras.

Contribution

It introduces a framework for duoidal structures on $ abla$-categories of operadic modules within $ abla$-categories, extending the theory of monoidal categories to a duoidal setting.

Findings

$ abla$-categories of operadic modules can be endowed with duoidal structures under certain conditions.

The paper characterizes when the category of $ abla$-$A$-modules admits a duoidal structure.

Provides new tools for studying algebraic structures in $ abla$-categories.

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 ${\rm Mod}_A^{\mathcal{O}}(\mathcal{C})$ 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 $A$.