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Hi, the page internally projective object of the nlab states that every projective object in a topos is internally projective. I also believed this to be true, but I found an exercise in “MacLane/Moerdijk — Sheaves in Geometry and Logic” whose phrasing suggests the contrary. The exercise is IV 16 (c) on page 217, it reads:
“In this part, assume that the terminal object $1\in\mathcal{E}$ is projective. Show that an internally projective object of $\mathcal{E}$ is projective. Is the converse also true? Show that every object of $\mathcal{E}$ is projective iff every object of $\mathcal{E}$ is internally projecitve.” (Boldface emphasize mine)
So if every projective were internally projective, then the universal quantification “every object of $\mathcal{E}$” would not be necessary.
Can anybody clarify this?
I think you are right and the page is wrong. At least, I can’t see right now any way to prove that projective => internally projective, although I don’t know a counterexample at the moment. My guess is that someone writing the page (perhaps me) wrongly extrapolated that from “all objects projective => all objects internally projective” (i.e. AC => IAC). Right now I think the correct extrapolation would be “stably projective => internally projective” (stably projective meaning its pullback to any slice category is projective).
I do know that the converse implication “internally projective => projective” is definitely only true under the assumption (stated by ML&M) that 1 is projective, since there are toposes which satisfy IAC but violate AC.
Thanks for your reply, Michael. I searched a bit more, and found a relevant lemma in the Elephant. It is Lemma D4.5.3, and it states:
“For an object $A$ of a topos $\mathcal{E}$, the following are equivalent:
$A$ is internally projective [i.e. $\Pi_A$ preserves epis].
$(-)^A:$ $\mathcal{E}$ $\to$ $\mathcal{E}$ preserves epis.
For every epimorphism $e:B$ $\rightarrow$ $A$, there exists $C$ with global support such that $C^*(e)$ is split epic.”
It is easy that 1 and 2 are equivalent, and 3 is implied by projectiveness of $A$, which would give an affirmative answer to the claim on the nlab page. However, I am not convinced that condition 3 is really strong enough to imply internal projectiveness, at least I don’t understand the proof. Johnstone writes:
“3=>1 since $C^*$ commutes with the functors $\Pi_A$ in an appropriate sense, and reflects epimorphisms.”
I read this as follows.
“ We want to show that $\Pi_A$ preserves epimorphisms. To this end, consider an epi $f:B$ $\to$ $A$ and let $C$ with global support such that $C^*e$ is split. Now to see that $\Pi_A e$ is epic, it suffices to show that $C^*\Pi_A e$ is epic, which follows from the stated commutation and the fact that $C^*e$ is split. ”
However, this reasoning establishes only that $\Pi_A$ preserves objects with global support, not that it preserves epis.
Am I overlooking something?
And has the property “stably projective” that you mention being studied? Google doesn’t give any results.
Ah, that result must have been what the author of that page was looking at. But it seems to me right now that you’re right: you need (iii) to remain true locally, i.e. for any epimorphism $e\colon B\to U\times A$, there should be an epic $p\colon C\to U$ such that $p^*(e)$ is split. Maybe that’s an error in the Elephant? He only uses (i) => (iii) in the proof of the subsequent corollary.
As for “stably projective”, I just made it up on the spur of the moment. (-:
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