Using the Quillen lifting property to define dividing lines and topological properties

A number of basic notions can be concisely and uniformly definedstarting from a list of (counter)examples using the lifting property,a simple category-theoretic tool used in a prominent way by Quillento axiomatise homotopy theory.As a warm-up, we shall showhow to reformulate the notions of contractibility and compactnessonly in terms the lifting property and (counter)exampleswhich are maps of finite topological spaces, usingthe (usual) category of topological spaces. We will alsostate conjectures/questions in point set topology these reformulations lead to.In the model theoretic part, we will show how to reformulate in this wayseveral no-order- and no-tree- propertiesusing generalised Stone spacesof models in a category of generalised topological spaces we define.The reformulated properties includeNOP, NTP, $NTP_i$, $NSOP_i$ (i⩾1), and NIP.Our reformulations raise the question whether Shelah's dividing lines areof homotopic nature, and suggest an approach how to devolop homotopy theoryfor model theory.