= Homotopy category
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The homotopy category of based CW complexes has spaces as objects and based homotopy classes as morphisms. Composition is well-defined because composition with continuous maps preserves homotopies. A based homotopy equivalence becomes an isomorphism. Rational localization instead inverts rational homotopy equivalences, producing a category represented by rational spaces. More general model categories have analogous homotopy categories obtained by inverting their weak equivalences.
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