  This paper gives a uniform-theoretic refinement of classical homotopy theory.
Both cubical sets (with connections) and uniform spaces admit classes of weak
equivalences, special cases of classical weak equivalences, appropriate for the
respective Lipschitz and uniform settings. Cubical sets and uniform spaces
admit the additional compatible structures of categories of (co)fibrant
objects. A categorical equivalence between classical homotopy categories of
cubical sets and spaces lifts to a full and faithful embedding from an
associated Lipschitz homotopy category of cubical sets into an associated
uniform homotopy category of uniform spaces. Bounded cubical cohomology
generalizes to a representable theory on the Lipschitz homotopy category.
Bounded singular cohomology on path-connected spaces generalizes to a
representable theory on the uniform homotopy category. Along the way, this
paper develops a cubical analogue of Kan's Ex^infinity functor and proves a
cubical approximation theorem for uniform maps.
