Strict n-categories and augmented directed complexes model homotopy types

TL;DR

This paper demonstrates that the homotopy categories of strict n-categories and augmented directed complexes are equivalent to the category of homotopy types, establishing a deep connection between these algebraic structures and topological spaces.

Contribution

The paper proves an abstract criterion for nerve functors to induce equivalences at the homotopy category level and applies it to strict n-categories and augmented directed complexes.

Findings

Homotopy categories of strict n-categories and augmented directed complexes are equivalent to homotopy types.

Established sufficient conditions for nerve functors to induce homotopy category equivalences.

Applied the abstract result to specific algebraic models, confirming their homotopy-theoretic relevance.

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.