From Nilspace Theory to Simplicial Homotopy Theory

We construct a faithful functor from the category of cubespaces, arising in structured ergodic theory and nilspace theory, to the category of (condensed) simplicial sets. This functor translates the weak structure theorem of a nilspace to a Postnikov tower of the corresponding simplicial set. In particular, nilspaces are mapped to Kan complexes and the structure groups of the nilspace correspond to the simplicial homotopy groups of the Kan complex. The functor is constructed via pullback along a special cosimplicial cubical set, whose combinatorial properties may be of independent interest.