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?



For which sites are all constant presheaves separated?
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.


