The uniform homotopy category
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Publication:6174700
DOI10.1016/J.JPAA.2023.107425zbMATH Open1524.18053arXiv2109.08576OpenAlexW3201254643MaRDI QIDQ6174700
Author name not available (Why is that?)
Publication date: 17 August 2023
Published in: (Search for Journal in Brave)
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.
Full work available at URL: https://arxiv.org/abs/2109.08576
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