# For which sites are all constant presheaves separated?

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 cheng.staff.shef.ac.uk/pssl88/pssl88-duncan.pdf 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 ncatlab.org/nlab/show/locally+connected+site –  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.