Group completion via the action $\infty$-category

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 $n\ge 2$. We will also show how this construction is related to a variety of other constructions of the group completion.

Revised and expanded Section 3.1 on actions and left modules. The original proposition 3.11 (v1) is replaced by several more granular statements. Additional minor fixes