  We illustrate the generative power of the lifting property (orthogonality of
morphisms in a category) as means of defining natural elementary mathematical
concepts by giving a number of examples in various categories, in particular
showing that many standard elementary notions of abstract topology can be
defined by applying the lifting property to simple morphisms of finite
topological spaces. Examples in topology include the notions of: compact,
discrete, connected, and totally disconnected spaces, dense image, induced
topology, and separation axioms. Examples in algebra include: finite groups
being nilpotent, solvable, torsion-free, p-groups, and prime-to-p groups;
injective and projective modules; injective, surjective, and split
homomorphisms.
  We include some speculations on the wider significance of this.
