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I've heard many times that for an algebraic topologist two spaces which are homotopy equivalent are essentially the same. But when the topological space is contractible then it is equivalent to a point. I wonder whether such situation can be put in the framework of the category theory-to be more precise: it is possible to define a category with objects being topological spaces but morphisms be defined in such a way that being isomorphic in this category is the same as being homotopy equivalent? Obviously such morphisms wouldn't be ordinary functions.

magma
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truebaran
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    The morphisms are homotopy classes of functions. The resulting category is called the homotopy category of spaces. This is standard material. – Qiaochu Yuan Apr 10 '14 at 03:42
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    @QiaochuYuan - that's the third answer truebaran has gotten, after Tyler's comment (now deleted) and mine :-) Yes, it's standard material, but the fact the morphisms can never be functions (implicit in the question) is a serious result. – theHigherGeometer Apr 10 '14 at 04:39
  • If you want to see an application of it that corresponds with what you are saying about contractible spaces then have a look at http://math.stackexchange.com/a/497780/75923 – drhab Apr 10 '14 at 18:15

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(Tyler answered in a comment after I started writing this. I'm posting it because of the extra information I give)

Yes-ish. I mean, you can do it, but for arbitrary topological spaces there is a difference between homotopy equivalence and weak homotopy equivalence, and you need to choose which one you care about.

The construction is well known, and goes as follows: take the category with objects topological spaces (or, for instance, those with the homotopy type of CW complexes, or other judicious choices, such as weak Hausdorff k-spaces) and the arrows are homotopy classes of continuous functions. That's it. An isomorphism in this category is a homotopy equivalence. Famously, the arrows of this category cannot in any way be faithfully represented as functions on sets (i.e. the category as defined here is not concrete).

theHigherGeometer
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