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Hmm, no doubt this is naïve, but the axiom as formulated there seems a bit fishy to me. I mean if the proposition does not make use of size in any way, it is perfectly reasonable that it can be resized up and down. But if the proposition does make use of size, it seems difficult to believe it can be resized down. Take for instance the example I mentioned in another thread, that any category with all (large) colimits is a poset. How does that resize down?
Edit: I suppose that this example cannot be formulated as a proposition in the intended sense?
I imagine your edit is correct. You’d need to express your proposed counterexample in HoTT as a certain type in $\mathcal{U}_{i+1}$.
Indeed. I’d be interested to know in a bit more detail why the example I mentioned is not a proposition in this sense, though (I imagine it is clear to an expert, but it is not to me!). I think it is clear that it lives in a higher universe than whichever universe one takes to correspond to small sets in the usual sense.
There is no “how” to resizing; it’s an axiom which baldly asserts that all propositions can be resized down. “Any category with all large colimits is a poset” is indeed a proposition and thus, assuming propositional resizing, can be resized down. The axiom therefore produces a small proposition that is equivalent to the large one, but it doesn’t tell you “what” that proposition is other than “the result of resizing this other one”.
Thanks! What I was getting at was that I do not know how to think about it, because naïvely it seems that some propositions might not be able to be sized down. In the HoTT book, though, it says that the axiom can be proven under the assumption of excluded middle, i.e. non-constructively. In other words, it must make sense. Maybe I can put it another way. Even if $Id(x,y)$ is inhabited for all $x, y : A$, $Id(x,y)$ might be too large to live in a universe $U$. So it would naïvely seem that there are clearly more propositions in $V \supset U$ than in $U$, unless $V$ is equivalent in size to $U$, which is impossible. It seems like one would somehow have to be able to ’discard’ all except $U$’s worth of the terms of $Id(x,y)$, and even with excluded middle that seems tricky to express.
Can you suggest why this is not the right way to think about it, and how one should rather think of it?
Edit: it might be enlightening to see the proof under the assumption of excluded middle.
Proof assuming LEM: given $P:Prop_{U_{i+1}}$, either $P\simeq 1$ or $P\simeq 0$ (that’s what LEM means). But we have $1:Prop_{U_i}$ and $0:Prop_{U_i}$, so in either case there is a $Q:Prop_{U_i}$ such that $P\simeq Q$.
It’s true that $\mathrm{Id}(x,y)$ may be too large to live in a universe $U$. That’s why the resizing axiom (as written here) states only that every proposition is some universe is equivalent (hence equal, assuming univalence) to a proposition in the smaller universe, not that it itself lives in a smaller one. A very “large” contractible type is nonetheless equivalent to a very small one. Voevodsky originally proposed a version of resizing as a rule stating that any proposition itself inhabits every universe:
$\frac{P:Prop_{U_{i+1}}}{P:Prop_{U_i}}$but to my knowledge it is not known whether this is consistent.
Ah, thanks! I see much better now. As you say, the key seems to be that the notion of equivalence is ’poly-universal’, whereas I was thinking in terms more like the stricter version of the rule that you mention.
Also I was thinking in more semantical terms, like I think one can if the proposition is ’semantically size-independent’, whereas this is not the right way to think about it, at least in general, here.
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