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1. • CommentRowNumber1.
   • CommentAuthorJonasFrey
   • CommentTimeAug 10th 2011
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   Author: JonasFrey
   Format: MarkdownItexHi, the page [internally projective object](http://ncatlab.org/nlab/show/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?

   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?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeAug 10th 2011
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   Author: Mike Shulman
   Format: MarkdownItexI 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorJonasFrey
   • CommentTimeAug 11th 2011
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   Author: JonasFrey
   Format: MarkdownItexThanks 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: 1. $A$ is internally projective [i.e. $\Pi_A$ preserves epis]. 2. $(-)^A:$ $\mathcal{E}$ $\to$ $\mathcal{E}$ preserves epis. 3. 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.

   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:

   1. $A$ is internally projective [i.e. $\Pi_A$ preserves epis].

   2. $(-)^A:$ $\mathcal{E}$ $\to$ $\mathcal{E}$ preserves epis.

   3. 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.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeAug 12th 2011
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   Author: Mike Shulman
   Format: MarkdownItexAh, 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. (-:

   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. (-: