Algebraic topology and homotopy in category theory

Question

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.

Answer 1

(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).