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    • CommentRowNumber1.
    • CommentAuthorJonasFrey
    • CommentTimeAug 10th 2011

    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 11\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?

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeAug 10th 2011

    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.

    • CommentRowNumber3.
    • CommentAuthorJonasFrey
    • CommentTimeAug 11th 2011

    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 AA of a topos \mathcal{E}, the following are equivalent:

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

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

    3. For every epimorphism e:Be:B \rightarrow AA, there exists CC with global support such that C *(e)C^*(e) is split epic.”

    It is easy that 1 and 2 are equivalent, and 3 is implied by projectiveness of AA, 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 *C^* commutes with the functors Π A\Pi_A in an appropriate sense, and reflects epimorphisms.”

    I read this as follows.

    “ We want to show that Π A\Pi_A preserves epimorphisms. To this end, consider an epi f:Bf:B \to A A and let CC with global support such that C *eC^*e is split. Now to see that Π Ae\Pi_A e is epic, it suffices to show that C *Π AeC^*\Pi_A e is epic, which follows from the stated commutation and the fact that C *eC^*e is split. ”

    However, this reasoning establishes only that Π A\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.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeAug 12th 2011

    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:BU×Ae\colon B\to U\times A, there should be an epic p:CUp\colon C\to U such that p *(e)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. (-: