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

The aim of this paper is to prove a generalization of the famous Theorem A of Quillen for strict $\mathrm{\infty }$-categories. This result is central to the homotopy theory of strict $\mathrm{\infty }$-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 $\mathrm{\infty }$-categorical methods.