Title: A Quillen's Theorem A for strict $\infty$-categories I: the simplicial proof,
Abstract: The aim of this paper is to prove a generalization of the famous Theorem A of
Quillen for strict $\infty$-categories. This result is central to the homotopy
theory of strict $\infty$-categories developed by the authors. The proof
presented here is of a simplicial nature and uses Steiner's theory of augmented
directed complexes. In a subsequent paper, we will prove the same result by
purely $\infty$-categorical methods.