We construct a functor associating a cubical set to a (simple) graph. We show that cubical sets arising in this way are Kan complexes, and that the A-groups of a graph coincide with the homotopy groups of the associated Kan complex. We use this to prove a conjecture of Babson, Barcelo, de Longueville, and Laubenbacher from 2006, and a strong version of the Hurewicz theorem in discrete homotopy theory.

We study cubical sets without degeneracies, which we call $\square$-sets. These sets arise naturally in a number of settings and they have a beautiful intrinsic geometry; in particular a $\square$-set $C$ has an infinite family of associated $\square$-sets $J^i(C)$, for $i=1,2,\ldots$, which we call James complexes. There are mock bundle projections $p_i \colon |J^i (C)| \to |C|$ (which we call James bundles) defining classes in unstable cohomotopy which generalise the classical James–Hopf invariants of $\Omega (S^2)$. The algebra of these classes mimics the algebra of the cohomotopy of $\Omega (S^2)$ and the reduction to cohomology defines a sequence of natural characteristic classes for a $\square$-set. An associated map to $BO$ leads to a generalised cohomology theory with geometric interpretation similar to that for Mahowald orientation.