# reference request – Topos with enough projective

It is often observed that each presposed topos has enough projectives, corollary of the result according to which representables are projective and each presumed is a colimit of representables. We also have that each The topos has enough injectives.

In this article, it is shown in the first two sections that a topos localic $$Sh (L)$$ has enough projectives if and only if $$L$$ has a base of objects "dislocatable".

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

I would like to know if there is a reference characterizing 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 topos of sheaves with defined values. Of course, a characterization of elementary topos would also be welcome.

I should note that it is a topos invariant more difficult to process than many, since the degenerate subpopies of any topos have enough projectives, so there is no larger Grothendieck topology guaranteeing a sufficient number of projectives. .