A topos may be thought of as a generalized topological space. Accordingly, the notions of
locally 2-connected space
etc.
have analogs for toposes and (∞,1)-toposes
locally connected topos
etc.
An object $A$ in a topos $\mathcal{E}$ is called a connected object if the hom-functor $\mathcal{E}(A, -)$ preserves finite coproducts.
Equivalently, an object $A$ is connected if it is nonempty (noninitial) and cannot be expressed as a coproduct of two nonempty subobjects.
A Grothendieck topos $\mathcal{E}$ is called a locally connected topos if every object $A \in \mathcal{E}$ is a coproduct of connected objects $\{A_i\}_{i \in I}$, $A = \coprod_{i \in I} A_i$.
It follows that the index set $I$ is unique up to isomorphism, and we write
This construction defines a functor $\Pi_0 : E \to Set : A \mapsto \pi_0(A)$ which is left adjoint to the constant sheaf functor, the left adjoint part of the global section geometric morphism.
Thus, for a locally connected topos we have
This is the connected component functor. It generalises the functor, also denoted $\pi_0$ or $\Pi_0$, which to a topological space assigns the set of connected components of that space. See the examples below.
The following proposition asserts that the existence of $\Pi_0$ already characterizes locally connected toposes.
A Grothendieck topos $\mathcal{E}$ is locally connected precisely if the global section geometric morphism $\Gamma : \mathcal{E} \to Set$ is an essential geometric morphism $(\Pi_0 \dashv L Const \dashv \Gamma) : \mathcal{E} \to Set$.
A proof appears as (Johnstone, lemma C.3.3.6).
Suppose that $(\Pi_0 \dashv L Const \dashv \Gamma) : \mathcal{E} \to Set$ exists.
First notice that an object $A$ is connected in the above sense precisely if $\Pi_0(A) = *$.
Because for all $S \in Set$ the connectivity condition demands that
but by the $(\Pi_0 \dashv L Const)$-hom-equivalence the first term is
and the last set is isomorphic to $S$ precisely for $\Pi_0(A)$ is the singleton set.
So we need to show that given the extra left adjoint $\Pi_0$, every object of $\mathcal{E}$ is a coproduct of objects for which $\Pi_0(-)$ is the point.
For that purpose consider for every object $A \in \mathcal{E}$ the pullback diagram
where the bottom morphism is the $(\Pi \dashv L Const)$-unit and the right isomorphism is the identification of any set as the colimit (here: coproduct) of the functor over the set itself that is constant on the point. Since pullbacks of isomorphism are isomorphisms, also the left morphism is an iso.
By universal colimits this left morphism is equivalently
and hence expresses $A$ as a coproduct of objects $i_A * *_s$, each of which is a pullback
where the right morphism includes the element $s$ into the set $\Pi_0 A$. By applying $\Pi_0$ to this diagram and pasting on the $(\Pi_0 \dashv L Const)$-counit we get
and by the zig-zag identity the bottom morphism is the identity. This says that in
all the component maps out of the coproduct factor through the point. This means that this can only be an isomorphism if all these component maps are point inclusions, hence if $\Pi_0(i_A^* *_s) \simeq *$ for all $s \in \Pi_0 A$.
However, this doesn’t mean that essential geometric morphisms are the “relative” analog of locally connected toposes; in general one needs to impose an additional condition, which is automatic in the case of the global sections morphism, to obtain the notion of a locally connected geometric morphism.
See at locally connected site.
For $C$ and $C$ cartesian closed categories, a functor $F : C \to D$ that preserves products is called a cartesian closed functor if the canonical natural transformation
(which is the adjunct of $F(A) \times F(B^A) \simeq F(A \times B^A) \to F(B)$) is an isomorphism.
The constant sheaf-functor $\Delta : \mathcal{S} \to \mathcal{E}$ is a cartesian closed functor precisely if $\mathcal{E}$ is a locally connected topos.
A topos $E$ is called a connected topos if the left adjoint $L Const : Set \to E$ is a full and faithful functor.
If $\Gamma \colon E\to Set$ is a locally connected topos, then it is also a connected topos — in that $L Const$ is full and faithful — if and only if the left adjoint $\Pi_0$ of $L Const$ preserves the terminal object.
This is (Johnstone, C3.3.3).
Notice that for a connected and locally connected topos, the adjunction
exhibits Set as a reflective subcategory of $E$. We may think then of Set as being the localization of $E$ at those morphisms that induce isomorphisms of connected components.
For $X$ a topological space, the category of sheaves $Sh(X) \coloneqq Sh(Op(X))$ is a locally connected topos precisely if $X$ is a locally connected space. The functor $\Pi_0$ sends a sheaf $F \in Sh(X)$ to the set of connected components of the corresponding etale space.
For $C =$ CartSp the site of Cartesian spaces with its good open cover coverage, the topos $Sh(CartSp)$ of smooth spaces is locally connected. An arbitrary $X \in Sh(CartSp)$ is sent to the colimit $\lim_\to X \in Set$. If $X$ is a diffeological space or even a smooth manifold, then this is the set of connected components of the underlying topological space.
Suppose that $C$ is a site such that constant presheaves on $C$ are sheaves. Then the left adjoint $\Pi_0$ exists and is given by the colimit functor: if we write $L : PSh(C) \to Sh(C)$ for sheafification, then for any sheaf $X$, we have
In particular, this is the case if every covering sieve in $C$ is connected, i.e. $C$ is a locally connected site.
If $C$ furthermore has a terminal object $1$, then the global sections functor $\Gamma\colon Sh(C)\to Set$ (the right adjoint of $L Const$) is simply given by evaluation at $1$, and so the unit $S \to \Gamma L Const S \cong L Const S(1)$ is an isomorphism. Thus in this case $Sh(C)$ is additionally connected. This situation also applies to $C=CartSp$ in example 2 above.
If $C$ is a category with all finite limits and if the unique functor $\pi \colon C \to \ast$ to the terminal category preserves covers (for $\ast$ equipped with the trivial topology/coverage) then $Sh(C)$ is locally connected. This is because the inclusion of the terminal object $i \colon \ast \to C$ provides a right adjoint to $\pi$, so that there is an adjoint quadruple of functors on presheaf categories
where $Lan_{(-)}$ and $Ran_{(-)}$ denote let and right Kan extension, respectively. Now if $C \to \ast$ indeed preserves covers and using that $C \to \ast$ trivially preserves finite limits and hence is a flat functor, then by the discussion at morphism of sites the first three functors here descend to sheaves and hence exhibit $Sh(C)$ as being locally connected.
But beware that the assumptions here are stronger than they may seem: that $C \to \ast$ preserves covers is not automatic, but is a strong condition. It is violated as soon as $C$ contains an empty object with empty cover, such as is the case in most categories of spaces, notably in categories of open subsets $Op(X)$ of a topological space $X$, as in example 1.
locally connected topos / locally ∞-connected (∞,1)-topos
Section C1.5 and C3.3 of
A variant is in
Discussion of characterizations of sites of definition of locally connected toposes is in