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

This paper is the second in a series of two papers about generalizing Quillen’s Theorem A to strict $\mathrm{\infty }$-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 $\mathrm{\infty }$-categorical nature of the same theorem. This proof is based on the theory of join and slices for strict $\mathrm{\infty }$-categories developed by the authors in a previous paper, and on a comma construction for strict $\mathrm{\infty }$-categories generalizing classical comma categories and Gray’s comma 2-categories. This $\mathrm{\infty }$-categorical comma construction is used by the first author in another paper to prove a generalization of Quillen’s Theorem B to strict $\mathrm{\infty }$-categories. We believe that the importance of this comma construction in the theory of $\mathrm{\infty }$-categories goes far beyond the scope of homotopy theory.