$\mathbb{E}_n$-algebras in m-categories

TL;DR

This paper establishes a connectivity bound for maps of $ $-operads and introduces an inductive method to construct $ $-algebras within $m$-categories, advancing the understanding of higher algebraic structures.

Contribution

It provides a new connectivity bound for $ $-operad maps and an inductive construction technique for $ $-algebras in $m$-categories, utilizing a novel Eckmann-Hilton argument.

Findings

Proved a connectivity bound for maps of $ $-operads.

Developed an inductive construction method for $ $-algebras in $m$-categories.

Established a Blakers-Massey type statement for algebras of coherent $ $-operads.

Abstract

We prove a connectivity bound for maps of $\infty$-operads of the form $\mathbb{A}_{k_1} \otimes \cdots \otimes \mathbb{A}_{k_n} \to \mathbb{E}_n$, and as a consequence, give an inductive way to construct $\mathbb{E}_n$-algebras in $m$-categories. The result follows from a version of Eckmann-Hilton argument that takes into account both connectivity and arity of $\infty$-operads. Along the way, we prove a technical Blakers-Massey type statement for algebras of coherent $\infty$-operads.