Homotopy MOC

Homotopy theory originated as a subfield of topology which is concerned with homotopy invariants rather than topological properties.

Originally this meant studying $𝖳 𝗈 𝗉 / ≃$ , where an object is a Topological space and a morphism is a homotopy class of continuous maps.
Focusing instead on Weak homotopy equivalence of continuous maps, we get the Homotopy category of topological spaces as the corresponding Homotopy category. This is equivalent to the full subcategory of $𝖳 𝗈 𝗉 / ≃$ spanned by CW complexes by the Cellular approximation theorem.
The homotopy theory described in ^2 is really about an induced (1,∞)-category, the homotopy category being a truncation of this. Thus we can generalize homotopy theory to the study of $(1, \infty)$ -categories (Higher category theory MOC).
A $(1, \infty)$ -category can be presented by a Category with weak equivalences. It is nicer when this presenting category has the structure of a Model category.

Homotopy theory of topological spaces

A Homotopy of continuous maps between maps preserves
A homotopy invariant is preserved by homotopy equivalences, i.e. shared by all homotopy equivalent spaces.
A CW complex is a nice topological space.