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.
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
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