Homotopy category

The naïve homotopy category $𝖳 𝗈 𝗉 / ≃$ is a quotient category built from $𝖳 𝗈 𝗉$ where objects remain topological spaces but hom-sets are homotopy quotients, i.e. morphisms are not continuous functions but homotopy classes of continuous functions. Topological spaces that are isomorphic in $𝖳 𝗈 𝗉 / ≃$ are called (strongly) homotopy equivalent.

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