Group completion via the action $\infty$-category

TL;DR

This paper generalizes Quillen's group completion construction to $E_n$-monoids within $ infty$-categories, providing a new model for group completion applicable for all $n \

Contribution

It introduces a novel $E_{n-1}$-monoidal $ infty$-category framework for group completion of $E_n$-monoids, extending classical results.

Findings

Provides a new model for group completion for $E_n$-monoids

Establishes the relation between the new construction and existing methods

Shows the construction models the group completion for $n \\geq 2$

Abstract

We give a generalization of Quillen's $S^{-1}S$ construction for arbitrary $E_n$-monoids as an $E_{n-1}$-monoidal $\infty$-category and show that its realization models the group completion provided that $n \geq 2$. We will also show how this construction is related to a variety of other constructions of the group completion.