On duoidal $\infty$-categories
Takeshi Torii
TL;DR
The work extends duoidal category theory to the ∞-categorical realm by introducing duoidal ∞-categories with two monoidal structures $\boxtimes$ and $\otimes$ linked via (op)lax coherence. It develops three parallel formulations of duoidal ∞-categories through bilax, double lax, and double oplax functors, yielding three corresponding ∞-categories and a robust theory of bimonoids, double monoids, and double comonoids. Central results show natural equivalences among algebraic structures: bimonoids in a duoidal ∞-category correspond to algebras of coalgebras and to coalgebras of algebras, realized through functors like $\text{Alg}$ and ${}_{}\text{cAlg}^{\otimes}$; analogous statements hold for double monoids/comonoids. The paper also develops mixed fibrations and monoid-object constructions in slice categories to support these equivalences and outlines higher-dimensional generalizations and concrete future examples, such as operadic modules and map monoidales in monoidal ∞-bicategories.
Abstract
A duoidal category is a category equipped with two monoidal structures in which one is (op)lax monoidal with respect to the other. In this paper we introduce duoidal $\infty$-categories which are counterparts of duoidal categories in the setting of $\infty$-categories. There are three kinds of functors between duoidal $\infty$-categories, which are called bilax, double lax, and double oplax monoidal functors. We make three formulations of $\infty$-categories of duoidal $\infty$-categories according to which functors we take. Furthermore, corresponding to the three kinds of functors, we define bimonoids, double monoids, and double comonoids in duoidal $\infty$-categories.