Discrete homotopy hypothesis for n‐types

Abstract We show that discrete and classical homotopy theories are equivalent after localizing at ‐equivalences for any non‐negative integer . By constructing an explicit homotopy inverse to the graph nerve functor associating an ‐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.