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Hyperconnected geometric morphisms
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”.
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 .
This includes for instance the case when 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.
This is further amplified by the following proposition. Recall that the inclusion Topos is reflective: it has a left adjoint: the localic reflection
Hyperconnected toposes are precisely those whose localic reflection is the point: .
Suppose is hyperconnected. Let be a locale and 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 fitting into this diagram
This establishes the natural equivalence
and hence identifies the point as the localic reflection of .
Conversely, suppose that has as localic reflection the point. The unit of the -adjunction – the reflector – is by essential uniqueness the global section geometric morphism.
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 as a localic topos for some locale , we have by the universal property of the adjunction unit an essentially unique lift in the left square
By the universal property of the pullback, this is then also an essentially unique solution to the original lifting problem.