  We define a variety of notions of cubical sets, based on sites organized
using substructural algebraic theories presenting PRO(P)s or Lawvere theories.
We prove that all our sites are test categories in the sense of Grothendieck,
meaning that the corresponding presheaf categories of cubical sets model
classical homotopy theory. We delineate exactly which ones are even strict test
categories, meaning that products of cubical sets correspond to products of
homotopy types.
