[dbranes]

The category of sets embeds into the category of topological spaces as a full subcategory, the essential image of which are the discrete spaces. Hence the equivalence I claimed is a precise statement.

Look up ring localization.

[NotAnEconomist]

> Hence the equivalence I claimed is a precise statement

They're obviously equivalent.

My point is that what's easily noticeable in one incarnation of the theory is different than what's easily noticeable in the other incarnation (or if you prefer, expressible), and switching our language for the same abstract structure can highlight different interesting features of it. And further, there's still utility to using topological perspectives and language to discuss the integers or naturals, even if it's equivalent to set theory.

I do appreciate the reference to ring localization -- will have to look at that further.

[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.)

[dbranes]

(the embedding is full & faithful)