[dbranes]

Genuinely tried reading the "examples" you wrote in this thread, can't make any sense out of it. Happy to discuss it if you clarify what you mean.

Just to address your original comment in this thread, perhaps it's relevant to note the following. Consider the homotopy theory of the category of nice topological spaces. The full subcategory of topological spaces supported on discrete topological spaces inherits a homotopy theory. This inherited homotopy theory is equivalent to the trivial homotopy theory on the category of sets: where weak equivalences are isomorphisms. This is the sense in which discrete spaces don't have an interesting homotopy theory, at least naively.

(This statement you can precise in your favorite model for the homotopy theory of spaces, via infinity categories, model categories etc.)