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It is often observed that every presheaf topos has enough projectives, as a corollary of the result that representables are projective and every presheaf is a colimit of representables. We also have that every topos has enough injectives.

In this paper, it is shown over the course of the first two sections that a localic topos $Sh(L)$ has enough projectives if and only if $L$ has a base of 'dislocable' objects.

On the other hand, there are certainly sheaf toposes without enough projectives: the references in this question suggest some examples even amongst toposes of sheaves on topological spaces.

I would like to know whether there exists a reference characterising when $Sh(\mathcal{C},J)$ has enough projectives. Note that I am not looking for projectives in any internal category of modules, but in the set-valued sheaf topos. Of course, a characterisation for elementary toposes would be equally welcome.

I should note that this is a harder invariant of toposes to deal with than many, since the degenerate subtopos of any topos trivially has enough projectives, so there is no largest Grothendieck topology which ensures having enough projectives.


Update: Having recently rediscovered an interest in this question, I obtained the following result.

Lemma: A Grothendieck topos $\mathcal{E}$ has enough projectives if and only if it has a separating set of projective objects.
Proof: Consider an arbitrary separating set of objects $H$; each object of $H$ admits a jointly epic family of morphisms from projective objects by assumption. Since a coproduct of projective objects is projective (which we can verify directly from the universal properties of coproducts and projectives), each such family can be taken to contain just a single object, and this set $G$ of projective covers of objects in $H$ is a separating set for the topos. $\square$

This suggests that it may be possible to obtain a site-theoretic characterisation of toposes with enough projectives after all.

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  • $\begingroup$ A larger class of topos that will satisfies this condition, is the toposes that are locally of the forms described in the paper. This corresponds exactly to toposes of equivariant sheaves over étale localic groupoid whose space of objects satisfies this condition of having a basis of dislocalbe objects. Similar results in topos theory suggest that this might be all of them, but I'm not quite sure yet. $\endgroup$ – Simon Henry Dec 1 '18 at 17:35
  • $\begingroup$ Hum, in second thought, there is also all presheaves toposes that have enough projective, and they are not all of the form I mentions above, so things are more complicated... $\endgroup$ – Simon Henry Dec 1 '18 at 17:49
  • $\begingroup$ Agreed: localic toposes provide good intuition generally, but since many toposes with enough projectives (especially presheaf toposes) are far from being determined by their subterminal objects, these are too restricted to be the whole story. $\endgroup$ – Morgan Rogers Dec 1 '18 at 18:22
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    $\begingroup$ Note that the first class I mentioned (the topos that are only locally of the form in the paper) is considerably larger than the localic examples and contains lots of example that are not "determined by their sub-terminal objects". For example all presheaves categories over an indexing category that has only monomorphisms fits into this class. $\endgroup$ – Simon Henry Dec 1 '18 at 21:49
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    $\begingroup$ Note also that there are subtleties lurking here -- for instance the distinction between a projective object and an internally projective object in a topos. $\endgroup$ – Tim Campion Dec 6 '18 at 14:12

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