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A homotopy category is a category whose objects are topological spaces and whose morphisms are homotopy classes of continuous functions. I wonder why the objects are spaces, instead of homotopy classes of spaces? Does the category whose objects are homotopy classes of spaces and whose morphisms are homotopy classes of continuous functions make sense? If we compare these two definitions, what are their differences and similarities?

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    $\begingroup$ You're probably interested in this question and this question from MO. Indeed what you want to to is identify the isomorphic objects in $hTop$. $\endgroup$ – Najib Idrissi May 19 '14 at 13:41
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    $\begingroup$ You're talking about replacing the homotopy category with its skeleton (en.wikipedia.org/wiki/Skeleton_(category_theory)). You should in general avoid doing this kind of thing as it requires making choices (to define homs you need to pick a representative of each homotopy class), and in more sophisticated contexts those choices can't be guaranteed to play nice with extra structure. $\endgroup$ – Qiaochu Yuan May 19 '14 at 17:18

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