Could not include topos theory - contents
A geometric morphism is hyperconnected if it is (left) orthogonal to a localic geometric morphism.
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”.
A geometric morphism $f\colon E\to F$ between toposes is called hyperconnected if the inverse image functor $f^*\colon F\to E$
its image is closed under subquotients in $E$.
This appears (Johnstone, p. 225).
If $g\colon C\to D$ is a functor between small categories which is both essentially surjective and full, then the induced geometric morphism $Set^C\to Set^D$ is hyperconnected. In fact, instead of essentially surjective it suffices for $g$ to be Cauchy surjective?, i.e. $D$ is the closure of $g(C)$ under retracts.
In particular, the global sections geometric morphism $Set^C\to Set$ on the presheaf topos is hyperconnected iff the category $C$ is strongly connected strongly connected (Johnstone, A4.6.9), i.e., inhabited and for any two objects $A,B$ there exist morphisms $f:A\to B$ and $g:B\to A$.
This includes for instance the case when $C=M$ is a monoid, and the topos of simplicial sets.
A locally connected and local topos is hyperconnected precisely if as a cohesive topos it satisfies pieces have points or equivalently discrete objects are concrete . See cohesive topos for details.
Any hyperconnected geometric morphism is connected,
So the name is not unreasonable.
Hyperconnected geometric morphisms are the left class of a 2-categorical orthogonal factorization system on the 2-category of toposes; the right class is the class of localic geometric morphisms.
See (Johnstone).
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.
This is further amplified by the following proposition. Recall that the inclusion $Sh(-) : Locale \simeq LocTopos \hookrightarrow$ Topos is reflective: it has a left adjoint: the localic reflection
Hyperconnected toposes $\mathcal{E}$ are precisely those whose localic reflection is the point: $L\mathcal{E} \simeq Sh(*) \simeq Set$.
Suppose $\Gamma : \mathcal{E} \to Set$ is hyperconnected. Let $X$ be a locale and $\mathcal{E} \to Sh(X)$ a geometric morphism. Notice that this sits in an essentially unique diagram
in Topos, where the vertical morphisms are the essentially unique global section geometric morphisms (we notationally suppress 2-isomorophisms).
By the above proposition there is an essentially unique geometric morphism $p : Set \to Sh(X)$ fitting into this diagram
This establishes the natural equivalence
and hence identifies the point as the localic reflection of $\mathcal{E}$.
Conversely, suppose that $\mathcal{E}$ has as localic reflection the point. The unit of the $(L \dashv Sh(-))$-adjunction – the reflector – $\mathcal{E} \to Set$ is by essential uniqueness the global section geometric morphism.
Let then
be a lifting problem, with the right morphism a localic geometric morphism. Since these are preserved by pullback in Topos, this is equivalently a diagram
Since this exhibits $f^* \mathcal{S}$ as a localic topos $Sh(X)$ for some locale $X$, we have by the universal property of the adjunction unit an essentially unique lift $p$ in the left square
By the universal property of the pullback, this is then also an essentially unique solution to the original lifting problem.