The Uniform Homotopy Category

TL;DR

This paper develops a uniform-theoretic refinement of classical homotopy theory, introducing new structures and equivalences for cubical sets and uniform spaces, and establishing categorical embeddings and cohomology theories in this refined setting.

Contribution

It introduces a uniform homotopy category, extends classical homotopy concepts to Lipschitz and uniform contexts, and constructs new cubical and cohomological tools.

Findings

Categorical equivalence between classical and uniform homotopy categories.

Bounded cubical cohomology and singular cohomology are representable in the new framework.

Develops a cubical analogue of Kan's Ex^infinity functor and proves a cubical approximation theorem.

Abstract

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.