Map monoidales and duoidal $\infty$-categories

TL;DR

This paper constructs duoidal $ abla$-categories within the framework of $ abla$-categories, introducing map $ abla$-monoidales in $ abla$-monoidal $( abla,2)$-categories, and explores their endomorphism $ abla$-categories with convolution products.

Contribution

It provides the first example of duoidal $ abla$-categories and develops a theory of map $ abla$-monoidales and their endomorphisms in $ abla$-monoidal $( abla,2)$-categories.

Findings

Endomorphism $ abla$-categories are coCartesian duoidal $ abla$-categories.

Convolution product on mapping $ abla$-categories is equivalent to the $ abla$-monoidal structure.

Introduces a new framework for duoidal structures in $ abla$-categories.

Abstract

In this paper we give an example of duoidal $\infty$-categories. We introduce map $\mathcal{O}$-monoidales in an $\mathcal{O}$-monoidal $(\infty,2)$-category for an $\infty$-operad $\mathcal{O}^{\otimes}$. We show that the endomorphism mapping $\infty$-category of a map $\mathcal{O}$-monoidale is a coCartesian $(\Delta^{\rm op},\mathcal{O})$-duoidal $\infty$-category. After that, we introduce a convolution product on the mapping $\infty$-category from an $\mathcal{O}$-comonoidale to an $\mathcal{O}$-monoidale. We show that the $\mathcal{O}$-monoidal structure on the duoidal endomorphism mapping $\infty$-category of a map $\mathcal{O}$-monoidale is equivalent to the convolution product on the mapping $\infty$-category from the dual $\mathcal{O}$-comonoidale to the map $\mathcal{O}$-monoidale.