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

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.

123 pages, in French, v2: revised according to the excellent suggestions of the referee, introduction partially rewritten, Section 7 slightly reorganized (see in particular Theorem 7.7 and Remark 7.9), first part of Appendix A rewritten, final version