  This paper is part of a series of papers about homotopy theory of strict
$n$-categories. In the first paper of this series, we gave conditions that
guarantee the existence of a Thomason model category structure on the category
of strict $n$-categories. The main goal of our paper is to show one of these
conditions. To do so, we associate to any simplicial complex a strict
$\infty$-category generated by a computad. We conjecture that this
$\infty$-category has the same homotopy type as the corresponding simplicial
complex and we prove this conjecture when the simplicial complex comes from a
poset. We introduce the notion of a quasi-initial object of an
$\infty$-category and we show that Street's orientals admit such an object. One
of the main tools used in this text is Steiner's theory of augmented directed
complexes.
