The “classical homotopy category” $Ho(Top)$ typically refers to the category of topological spaces with morphisms between them the homotopy classes of continuous functions, or (slightly less classically but more commonly these days) to its full subcategory on those topological spaces homeomorphic to a CW-complex. The latter is technically the homotopy category obtained by localizing the category of topological spaces at those continuous functions that are weak homotopy equivalences, hence it is also the homotopy category of a model category of the classical model structure on topological spaces.
By $Ho(Top)$ one denotes the category which is the homotopy category of Top with respect to weak equivalences given
either by homotopy equivalences – $Ho(Top)_{he}$.
or by weak homotopy equivalences – $Ho(Top)_{whe}$.
Depending on context here Top contains all topological spaces or is some subcategory of nice topological spaces.
The study of $Ho(Top)$ was the motivating example of homotopy theory. Often $Ho(Top)$ is called the homotopy category.
The simplicial localization of Top at the weak homotopy equivalences yields the (∞,1)-category of ∞-groupoids/homotopy types.
Let now $Top$ denote concretely the category of compactly generated weakly Hausdorff spaces. And Let $CW$ be the subcategory on CW-complexes. We have $Ho(CW)_{whe} = Ho(CW)_{he} = Ho(CW)$.
There is a functor
that sends each topological space to a weakly homotopy equivalent CW-complex.
By the homotopy hypothesis-theorem $Ho(CW)$ is equivalent for instance to the homotopy category of a model category $Ho(sSet_{Quillen})$ of the classical model structure on simplicial sets as well as $Ho(Top_{Quillen})$of the classical model structure on topological spaces.
The category $Ho(Top)_{he}$ can be studied by testing its objects with objects from $Ho(CW)$. This is the topic of shape theory.