On duoidal \(\infty \)-categories

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.