More generally, a connected geometric morphism is a “relativized” notion of this, saying that is “connected as a topos over .”
A Grothendieck topos is connected if the unique geometric morphism is connected. If is the topos of sheaves on a topological space (or more generally a locale), then this is equivalent to the usual definition of connectedness for (see C1.5.7 in the Elephant).
For geometric morphisms which are also locally connected, connectedness can be phrased in an especially nice form.
If is locally connected, then it is connected if and only if the left adjoint of the inverse image functor (which exists, since is locally connected) preserves the terminal object.
On the one hand, if is fully faithful, then the counit is an isomorphism, so we have ; hence preserves the terminal object.
On the other hand, suppose that preserves the terminal object. Suppose also for simplicity that . Then any set is the coproduct of copies of the terminal object. But and both preserve coproducts (since they are left adjoints) and terminal objects (since is left exact, and by assumption for ), so we have
Thus, the counit is an isomorphism, so is fully faithful.
When is not , we just have to replace ordinary coproducts with “-indexed coproducts,” regarding and as -indexed categories.
(This is C3.3.3 in the Elephant.)
Strengthenings of this condition include
a strongly connected geometric morphism, which is locally connected such that preserves all finite products, and
a totally connected geometric morphism, which is locally connected such that preserves all finite limits.
As explained at locally connected site, when is locally connected, the left adjoint is simply obtained by taking colimits over . Now by the co-Yoneda lemma, the colimit over any representable presheaf is a singleton (i.e. a terminal object in Set):
But if has a terminal object, then that terminal object represents the terminal presheaf, which is also the terminal presheaf. Hence under these conditions, preserves the terminal object, so is connected.
Since the functor sending a topos to itself and a geometric morphism to its inverse image functor is 2-fully-faithful (an equivalence on hom-categories), connected morphisms are representably co-fully-faithful in .
Therefore, for 2-categorical orthogonality it suffices to show that in any commutative (up to iso) square
of geometric morphisms in which is connected and is etale, there exists a filler such that and .
However, if is such that (such exists by definition of being etale), then for any topos equipped with a geometric morphism , lifts of along are equivalent to morphisms in . In particular, is determined by a map , and since and is fully faithful, this map comes from a map in , which in turn determines a geometric morphism which is the desired filler.
Any locally connected geometric morphism factors as a connected and locally connected geometric morphism followed by an etale one.
Given locally connected, we can factor it as . The second map is etale by definition, while the first is locally connected (the left adjoint is essentially again) and connected since it preserves the terminal object (by construction).
(Connected, Etale) is a factorization system on the 2-category of toposes and locally connected geometric morphisms.
The category of etale geometric morphisms over a base topos , which is equivalent to itself, is a reflective subcategory of the slice 2-category . The reflector constructs “ of a locally connected topos.”
These results all have generalizations to ∞-connected (∞,1)-toposes.
The gros sheaf topos on the site CartSp – which contains the quasi-topos of diffeological space – is a connected topos, since the site CartSp is a locally connected site and contains a terminal object.
For every object , we have that sits over by the etale geometric morphism.
This makes be a locally connected topos.
Notice that the terminal object of is . If now is connected, then
and so the extra left adjoint preserves the terminal object. By the above proposition this means that is also connected.
connected topos / ∞-connected (∞,1)-topos