Strict n-categories and augmented directed complexes model homotopy types

Abstract

In this paper we show that both the homotopy category of strict $n$-categories, $1\leqslant n \leqslant \infty$, and the homotopy category of Steiner's augmented directed complexes are equivalent to the category of homotopy types. In order to do so, we first prove an abstract result, based on a strategy of Fritsch and Latch, giving sufficient conditions for a nerve functor with values in simplicial sets to induce an equivalence at the level of homotopy categories. We then apply this result to strict $n$-categories and augmented directed complexes, for which the hypothesis of our theorem were established by Ara and Maltsiniotis.