On higher monoidal $\infty$-categories

TL;DR

This paper introduces a generalized framework for higher monoidal $5$-categories using $5$-operads, establishing an equivalence between coCartesian and Cartesian $5$-monoidal categories with reversed operad sequences.

Contribution

It generalizes the concept of higher monoidal categories in $5$-categories by defining $5$-monoidal $5$-categories for sequences of $5$-operads and proves an equivalence involving reversed sequences.

Findings

Established an equivalence between coCartesian and Cartesian $5$-monoidal categories with reversed operad sequences.

Generalized higher monoidal categories to sequences of $5$-operads.

Provided a duality framework relating different types of $5$-monoidal $5$-categories.

Abstract

In this paper we introduce a notion of $\mathbf{O}$-monoidal $\infty$-categories for a finite sequence $\mathbf{O}^{\otimes}$ of $\infty$-operads, which is a generalization of the notion of higher monoidal categories in the setting of $\infty$-categories. We show that the $\infty$-category of coCartesian $\mathbf{O}$-monoidal $\infty$-categories and right adjoint lax $\mathbf{O}$-monoidal functors is equivalent to the opposite of the $\infty$-category of Cartesian $\mathbf{O}_{\rm rev}$-monoidal $\infty$-categories and left adjoint oplax $\mathbf{O}_{\rm rev}$-monoidal functors, where $\mathbf{O}^{\otimes}_{\rm rev}$ is a sequence obtained by reversing the order of $\mathbf{O}^{\otimes}$.