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Lemma D4.5.3 of Johnstone’s Sketches of an Elephant states:

Lemma. For an object $A$ of a topos $\newcommand{\E}{\mathcal{E}}\E$, the following are equivalent:

  1. $A$ is internally projective [i.e. $\Pi_A : \E/A \to \E$ preserves epimorphisms];
  2. $(−)^A : \E \to \E$ preserves epimorphisms;
  3. for every epimorphism $\newcommand{\epito}{\twoheadrightarrow} e : B \epito A$ in $\E$, there exists $C \epito 1$ such that $C^∗(e)$ is split epic.

(These conditions should be (i), (ii), (iii), but lettered lists don’t seem to be available in this Markdown dialect.)

The implications (i) $\newcommand{\Iff}{\Leftrightarrow} \Iff$ (ii) and (ii) $\newcommand{\Imp}{\Rightarrow}\Imp$ (iii) are fine, but (iii) $\Imp$ (i) is rather murky. The argument given in the Elephant is a very brief sketch; I and a couple of colleagues spent some time today trying to figure out the details, couldn’t, and ended up doubting that this implication is correct. Searching around, this nforum thread shows that Mike Shulman and Jonas Frey have previously come to essentially the same conclusion.

However, we couldn’t find a counterexample, and the nforum thread doesn’t give one either, so the question is still a bit unsettled. Does anyone know either a counterexample to the implication (iii) $\Imp$ (i), or else an argument that it holds?

To keep any discussion clear, I suggest the term weakly projective for condition (iii) (unless someone knows a more established term for it). As Mike says in the nforum thread, if you strengthen this to its stable version, by quantifying over epis $e : B \to U \times A$ for all objects $U$, then it does imply internal projectivity.

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There is a counterexample, due to Todd Trimble, in another nForum thread; cf. the nLab entry on internally projective objects. Theorem 2 of loc. cit. tells that for an object $A$ to be internally projective, it is necessary and sufficient that for all objects $B$ and $U$ and every epimorphism $e\colon B\to A\times U$ there is an epimorphism $f\colon C\to U$ such that the pull back morphism $(\operatorname{id}_A\times f)^*(e)\colon (\operatorname{id}_A\times f)^*B\to A\times C$ is split. Right after the proof it is noted that in Lemma 4.5.3 of the Elephant, the third statement is just the the special case $U=1$ and that this is insufficient to prove that $A$ is internally projective,

[...] since if so, then every projective object would be internally projective, which as we show below is not the case.

In fact, this is not "shown below", but a link is provided to this counterexample (which is just the same as the one in the aforementioned nForum entry by Todd Trimble). Note that by proposition 1 of the nLab entry, if the topos has enough projectives and projectives are closed under binary products, then every projective object is internally projective.

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