In particular, a hyperconnected topos is a topos that is “as far from being a localic topos as possible”. In view of the fact that a topos is a generalized space, while a localic topos is an ordinary topological space/locale, regarded as a topos, this means that hyperconnected toposes are the “purely-generalized generalized spaces”.
This appears (Johnstone, p. 225).
If is a functor between small categories which is both essentially surjective and full, then the induced geometric morphism is hyperconnected. In fact, instead of essentially surjective it suffices for to be Cauchy surjective?, i.e. is the closure of under retracts.
In particular, the global sections geometric morphism on the presheaf topos is hyperconnected iff the category is strongly connected strongly connected (Johnstone, A4.6.9), i.e., inhabited and for any two objects there exist morphisms and .
Any hyperconnected geometric morphism is connected,
So the name is not unreasonable.
In particular, a geometric morphism can only be both hyperconnected and localic if it is an equivalence. Therefore, if we view topoi as generalized topological spaces (or locales), the world of hyperconnected topoi and geometric morphisms lives entirely in the “generalized” part.
Hyperconnected toposes are precisely those whose localic reflection is the point: .
By the above proposition there is an essentially unique geometric morphism fitting into this diagram
This establishes the natural equivalence
and hence identifies the point as the localic reflection of .
By the universal property of the pullback, this is then also an essentially unique solution to the original lifting problem.