The homotopy type of the $\infty$-category associated to a simplicial complex

Abstract

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.