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Perhaps more interesting would be investigating a function whose value at $n$ is "the number of interesting examples of $n$-chotomies in mathematics" as I suspect it would decrease rather quickly. However, philosophically, there is a probably a good reason that there are many examples of dichotomies and trichotomies, as opposed to, say 11-chotomies. Perhaps this is connected to the relative smallness of the human mind, in some sense. Additionally though, such a philosophical argument would show that such a question is pretty silly.
Wow thanks very much Urs. I probably should have brought this up on the nForum to avoid questions of "usefulness" but you found it anyway! As usual, there's quite a bit to digest here, but I very much appreciate you taking the question seriously. I suspected that this question was not fully resolved. Exciting!
@Zhen there is a model structure on the category of sets, determined by inclusions and surjections, can we not come up with something in a more general topos, based on the subobject classifier? moreoever, as we can localize heyting algebras, can we not think of localizing as changing the subobject classifier possibly?
This is a fascinating question. One quick remark that is maybe way wrong: I was thinking last night (as I fell asleep, which is when I do some of my best and worst thinking) that maybe forcing could be thought of in terms of localizations? If we want to think of our set theory in terms of sheaves on a topos, can we think of forcing as being like localizing this category of sheaves (as it has a model category structure) and making certain maps "equivalences", i.e. making certain things "equivalent" to being true? Or something? I'm not much of a set-theorist...
Yeah @Fernando, the proof that there are not small objects is in an appendix of Hovey and Strickland's paper on localizations at Morava K-theories, the exact title of which escapes me at the moment. The basic idea is, the colocalizing category of $E$-local objects has small objects if and only if $\langle E\rangle\geq\langle F(n)\rangle$ for some finites type-n spectrum $F(n)$. He then proves that if $\langle E\rangle\geq\langle F(n)\rangle$ for some $n$ then $L_EI\neq 0$, where $I$ is the Brown-Comenetz dual of the sphere. However, localizing at $BP$, $I$, $HF_p$ or the harmonic spectrum..
