Discrete homotopy hypothesis for n-types

Abstract

We show that discrete and classical homotopy theories are equivalent after localizing at n-equivalences for any non-negative integer n. By constructing an explicit homotopy inverse to the graph nerve functor associating an n-fibrant cubical set to a graph, we are also able to give explicit computations of several previously unknown discrete homotopy groups of boundaries of cubes and suspensions of cycles.