  In the context of relative topos theory via stacks, we introduce the notion
of existential fibred site and of existential topos of such a site. These
notions allow us to develop relative topos theory in a way which naturally
generalizes the construction of toposes of sheaves on locales, and also
provides a framework for investigating the connections between Grothendieck
toposes as built from sites and elementary toposes as built from triposes. Then
we focus on fibred preorder sites and establish a fibred generalisation of the
ideal-completion of a preorder site. Lastly, we provide an explicit description
of the hyperconnected-localic factorization of a geometric morphism in terms of
internal locales.
