Corollary 2.3.7. The homotopy category \(\Ho (\An )\) is equivalent to the category \(\h \Cell \) whose objects are the cell complexes and whose morphism set between \(X\) and \(Y\) is the set \([X,Y]\) of homotopy classes of continuous maps \(X \to Y\).
Proof. It suffices to observe that a functor \(F\colon \Cell \to C\) into a 1-category inverts homotopy equivalences if and only if it identifies homotopic maps. One implication follows by applying \(F\) to homotopy inverses. Conversely, if \(H\colon f \sim g\) is a homotopy, then \[ F(f)=F(H)F(i_0)=F(H)F(\pr )^{-1}=F(H)F(i_1)=F(g), \] where \(i_0,i_1\colon X \to X\times [0,1]\) are the endpoint inclusions and \(\pr \colon X\times [0,1]\to X\) is the projection. Thus \(\h \Cell \) has the same universal property as the homotopy category of the localization in Proposition 2.3.6. □
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