A Quillen's Theorem A for strict $\infty$-categories II: the $\infty$-categorical proof

TL;DR

This paper provides a conceptual, $ abla$-categorical proof of Quillen's Theorem A for strict $ abla$-categories, utilizing join, slices, and a generalized comma construction, advancing the theoretical framework of $ abla$-categories.

Contribution

It introduces an $ abla$-categorical proof of Quillen's Theorem A, based on join, slices, and a new comma construction for strict $ abla$-categories.

Findings

A new $ abla$-categorical proof of Theorem A.

Development of a generalized comma construction for strict $ abla$-categories.

Demonstration of the importance of comma constructions beyond homotopy theory.

Abstract

This paper is the second in a series of two papers about generalizing Quillen's Theorem A to strict $\infty$-categories. In the first one, we presented a proof of this Theorem A of a simplicial nature, direct but somewhat ad hoc. In the current paper, we give a conceptual proof of an $\infty$-categorical nature of the same theorem. This proof is based on the theory of join and slices for strict $\infty$-categories developed by the authors in a previous paper, and on a comma construction for strict $\infty$-categories generalizing classical comma categories and Gray's comma 2-categories. This $\infty$-categorical comma construction is used by the first author in another paper to prove a generalization of Quillen's Theorem B to strict $\infty$-categories. We believe that the importance of this comma construction in the theory of $\infty$-categories goes far beyond the scope of homotopy theory.