### A Quillen's Theorem A for strict ∞-categories II: the ∞-categorical proof

#### Abstract

This paper is the second in a series of two papers about generalizing Quillen's Theorem A to strict ∞-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 ∞-categorical nature of the same theorem. This proof is based on the theory of joint and slices for strict ∞-categories developed by the authors in a previous paper, and on a comma construction for strict ∞-categories generalizing classical comma categories and Gray's comma 2-categories. This ∞-categorical comma construction will be used by the first author in a forthcoming paper to prove a generalization of Quillen's Theorem B to strict ∞-categories. We believe that the importance of this comma construction in the theory of ∞-categories goes far beyond the scope of homotopy theory.

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