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arXiv · 2610.20428

The classical homotopy category is cartesian closed

Zhen Lin Low
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Abstract

The category of all topological spaces and homotopy classes of maps is cartesian closed. In fact, the $(\infty, 1)$-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.

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Zhen Lin Low. 2026-10-14. The classical homotopy category is cartesian closed. https://arxiv.org/abs/2610.20428

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