typical contexts
If we view a (Grothendieck) topos as a generalized topological space, then a connected topos is a generalization of a connected topological space.
More generally, a connected geometric morphism $p\colon E\to F$ is a “relativized” notion of this, saying that $E$ is “connected as a topos over $F$.”
A geometric morphism $p\colon E\to F$ is connected if its inverse image part $p^*$ is full and faithful.
A Grothendieck topos $E$ is connected if the unique geometric morphism $E \to Set = Sh(*)$ is connected. If $E$ is the topos of sheaves on a topological space $X$ (or more generally a locale), then this is equivalent to the usual definition of connectedness for $X$ (see C1.5.7 in the Elephant).
Equivalently, a topos is connected if its global section geometric morphism exhibits discrete objects.
Connected geometric morphisms are in particular surjective.
For geometric morphisms which are also locally connected, connectedness can be phrased in an especially nice form.
If $p\colon E\to F$ is locally connected, then it is connected if and only if the left adjoint $p_!$ of the inverse image functor (which exists, since $p$ is locally connected) preserves the terminal object.
On the one hand, if $p^*$ is fully faithful, then the counit $p_! p^* \to \Id$ is an isomorphism, so we have $p_!(*) \cong p_!(p^*(*)) \cong *$; hence $p_!$ preserves the terminal object.
On the other hand, suppose that $p_!$ preserves the terminal object. Suppose also for simplicity that $F=Set$. Then any set $A$ is the coproduct $\coprod_A *$ of $A$ copies of the terminal object. But $p^*$ and $p_!$ both preserve coproducts (since they are left adjoints) and terminal objects (since $p^*$ is left exact, and by assumption for $p_!$), so we have
Thus, the counit $p_! p^* \to \Id$ is an isomorphism, so $p^*$ is fully faithful.
When $F$ is not $Set$, we just have to replace ordinary coproducts with “$F$-indexed coproducts,” regarding $E$ and $F$ as $F$-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 $p_!$ preserves all finite products, and
a totally connected geometric morphism, which is locally connected such that $p_!$ preserves all finite limits.
If $C$ is a locally connected site with a terminal object, then the topos of sheaves $Sh(C)$ on $C$ is (not just locally connected) but connected.
As explained at locally connected site, when $C$ is locally connected, the left adjoint $\Pi_0\colon Sh(C) \to Set$ is simply obtained by taking colimits over $C^{op}$. Now by the co-Yoneda lemma, the colimit over any representable presheaf is a singleton (i.e. a terminal object in Set):
But if $C$ has a terminal object, then that terminal object represents the terminal presheaf, which is also the terminal presheaf. Hence under these conditions, $\Pi_0$ preserves the terminal object, so $Sh(C)$ is connected.
Connected geometric morphisms are left orthogonal to etale geometric morphisms in the 2-category Topos.
Since the functor $Topos^{op} \to Cat$ 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 $Topos$.
Therefore, for 2-categorical orthogonality it suffices to show that in any commutative (up to iso) square
of geometric morphisms in which $p$ is connected and $q$ is etale, there exists a filler $h\colon C\to B$ such that $h p \cong f$ and $q h \cong g$.
However, if $X\in D$ is such that $B \cong D/X$ (such exists by definition of $q$ being etale), then for any topos $E$ equipped with a geometric morphism $k\colon E\to D$, lifts of $k$ along $q$ are equivalent to morphisms $* \to k^*(X)$ in $C$. In particular, $f$ is determined by a map $*\to f^*(q^*(X)) \cong p^*(g^*(X))$, and since $* \cong p^*(p_*(*))$ and $p^*$ is fully faithful, this map comes from a map $*\to g^*(X)$ in $C$, which in turn determines a geometric morphism $h\colon C\to B$ 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 $f\colon E\to S$ locally connected, we can factor it as $E \to S/f_!(*) \to S$. The second map is etale by definition, while the first is locally connected (the left adjoint is essentially $f_!$ again) and connected since it preserves the terminal object (by construction).
In particular:
(Connected, Etale) is a factorization system on the 2-category $LCTopos$ of toposes and locally connected geometric morphisms.
The category of etale geometric morphisms over a base topos $S$, which is equivalent to $S$ itself, is a reflective subcategory of the slice 2-category $LCTopos/S$. The reflector constructs “$\Pi_0$ of a locally connected topos.”
These results all have generalizations to ∞-connected (∞,1)-toposes.
The gros sheaf topos $Sh(CartSp)$ 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.
Let $\Gamma : \mathcal{E} \to Set$ be a connected and locally connected topos and $X \in \mathcal{E}$ a connected object, $\Pi_0(X) \simeq *$. Then the over-topos $\mathcal{E}/X$ is also connected and locally connected.
For every object $X$, we have that $\mathcal{E}/X$ sits over $\mathcal{E}$ by the etale geometric morphism.
This makes $\mathcal{E}/X$ be a locally connected topos.
Notice that the terminal object of $\mathcal{E}/X$ is $(X \stackrel{Id}{\to} X)$. If now $X$ is connected, then
and so the extra left adjoint $\Pi_0 \circ X_!$ preserves the terminal object. By the above proposition this means that $\mathcal{E}/X$ is also connected.
connected topos / ∞-connected (∞,1)-topos