PreprintarXiv licencearXiv
The classical homotopy category is cartesian closed
Abstract
The category of all topological spaces and homotopy classes of maps is cartesian closed. In fact, the (∞, 1)-category of all topological spaces and continuous maps, localised at homotopy equivalences, is cartesian closed. We model the exponential object using a topological space whose points are continuous maps equipped with putative neighbourhood filters. In general, this topological space will not even be T1-separated, but the evaluation map induces a natural weak homotopy equivalence witnessing the universal property.
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BibTeX
@misc{low2026classical,
title = {{The classical homotopy category is cartesian closed}},
author = {Zhen Lin Low},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20428},
eprint = {2610.20428},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20428},
}
RIS
TY - UNPB
TI - The classical homotopy category is cartesian closed
AU - Zhen Lin Low
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20428
UR - https://arxiv.org/abs/2610.20428
AB - The category of all topological spaces and homotopy classes of maps is cartesian closed. In fact, the (∞, 1)-category of all topological spaces and continuous maps, localised at homotopy equivalences, is cartesian closed. We model the exponential object using a topological space whose points are continuous maps equipped with putative neighbourhood filters. In general, this topological space will not even be T_1-separated, but the evaluation map induces a natural weak homotopy equivalence witnessing the universal property.
ER -