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I'm intererested in open surjective geometric morphisms induced by fibrations of sites $S\to T$ a la Moerdijk, but as a warm-up, let's consider the case $S \to \ast$. In the case that $S$ is a poset $P$ as naturally turns up in set theory, then every constant presheaf is separated. My naive extrapolation is that this is at least partly due to the fact $P$ has no initial object, so we don't have any alteration there when applying the functor $(-)^+$ to the constant presheaf.

But is this enough in general? Do we need a condition like asking that the topos $Sh(S)$ is localic?

And then I'd like to know the relative case: given a fibration of sites $S\to T$, when are presheaves on $S$ that are given by pullback along the fibration of sheaves on $T$ separated?

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See e.g. here for the definition of a fibration of sites, if you can't get the Elephant or Moerdijk's original paper. – David Roberts Jun 19 '13 at 11:55
Not sure about separated right now, but for actual sheaves, you need something like this here – Urs Schreiber Jun 19 '13 at 23:37
Hmm, it might be that I need the weaker condition noted under definition 1, that all covering families are inhabited. I saw this in the Elephant in relevant sections, but I didn't notice whether it gave me what I wanted. (This condition is in particular violated when we have the empty covering family of initial objects.) – David Roberts Jun 19 '13 at 23:49

OK, I've found the answer in the Elephant. Given the map of sites $p\colon S\to \ast$, we get a functor of toposes $Set = Pre(\ast) \to Pre(S)$ by precomposition with $p$, which lands in separated presheaves precisely when $p$ preserves covers. Since $\ast$ has only the cover $id\colon \ast \to \ast$, then we see that constant presheaves are separated if and only if all covering families are inhabited.

More generally, for the case of a fibration of sites, we need $S\to T$ to preserve covering families, which is equivalent to asking that $S\to T$ induces a geometric morphism which is an open surjection.

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Thanks to Urs for giving my the comment that catalysed finding this answer! – David Roberts Jun 20 '13 at 3:07
And I've added this as an example at – David Roberts Jun 20 '13 at 3:48
Thanks for adding this Example to the nLab page!! Could you also add either a pointer to the proof in the Elephant or else add the proof itself to the nLab page? Thanks! (Simple as it may be, but it's good practice to support the statements on the nLab with proof or with pointers to proofs. Nobody knows which background readers will have who come across it...) – Urs Schreiber Jun 20 '13 at 8:56
Will do later today... – David Roberts Jun 20 '13 at 23:18

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