Discrete homotopy theory, introduced by H. Barcelo and collaborators, is a homotopy theory of (simple) graphs. Homotopy invariants of graphs have found numerous applications, for instance, in the theory of matroids, hyperplane arrangements, topological data analysis, and time series analysis. Discrete homotopy theory is also a special instance of a homotopy theory of simplicial complexes, developed by R. Atkin, to study social and technological networks.

I will report on the joint work with D. Carranza (arXiv:2202.03516) on developing a new foundation for discrete homotopy theory, based on the homotopy theory of cubical sets. We use this foundation to prove the conjecture of Babson, Barcelo, de Longueville, and Laubenbacher from 2006 relating homotopy groups of a graph to the homotopy groups of a certain cubical complex associated to it, as well as a discrete homotopy theory analogue of the Hurewicz theorem.