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 $\mathcal{E}\big(A, -\big)$ preserves finite coproducts.
Equivalently, an object $A$ is connected if it is non-empty (in that it is not the initial object) 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 is a coproduct of connected objects (Def. ), hence if for $A \in \mathcal{E}$ there exists $\big\{A_i \in \mathcal{E}\big\}_{i \in I}$ such that
If this is the case, it follows that the index set $I$ is unique up to isomorphism, and we denote it by
This construction defines a functor
which is left adjoint to the locally constant sheaf functor, the left adjoint part of the global section geometric morphism.
This is the connected component functor. It generalizes 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.
In summary, for a locally connected topos the terminal geometric morphism extends to an adjoint triple of this form:
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 $(\Pi_0 \dashv L Const \dashv \Gamma) : \mathcal{E} \to Set$.
The “only if”-case was just claimed/argued above, we need to show the “if”-case.
Hence suppose that $\Pi_0$ with $(\Pi_0 \dashv LConst)$ exists. We will show that then every object is a coproduct of connected objects. (A proof also appears as (Johnstone, Lemma C.3.3.6).)
First we claim that an object $A$ is connected in the above sense precisely if $\Pi_0(A) = \ast$.
To see this, observe that
where we used that $LConst$ is a left adjoint and that left adjoints preserve colimits (hence preserve initial objects), we consideted the adjunction unit $\eta_A$, and where the last implication follows since the initial object in any topos is strict.
But this gives
because if $A$ as on the left were the coproduct of non-initial $A_1$, $A_2$, then also $\Pi_0(A)$ would be the coproduct of non-initial $\Pi_0(A_1)$, $\Pi_0(A_2)$, by (1), which would contradict the assumption that $\Pi_0(A) \simeq \ast$.
Conversely, to see
observe that the connectivity assumption implies in particular that
for all $S \in Set$, where in the first step we used the $\big(\Pi_0 \dashv LConst\big)$-hom isomorphism. But this says that $\Pi_0(A)$ is connected as an object of Sets, hence that it is the singleton set.
With this equivalence
in hand (given the extra left adjoint $\Pi_0$), we are now reduced to showing that every object of $\mathcal{E}$ is a coproduct of objects for which $\Pi_0(-)$ is the point.
For that purpose consider for $A \in \mathcal{E}$ the following 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 isomorphisms are preserved under pullback (here), it follows that also the left morphism is an isomorphism, as shown.
Now, by the fact that a topos has “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 $s \in \Pi_0 A$.
Beware that Prop. only applies to terminal geometric morphisms (of locally connected toposes). The analogous “relative” characterization of more general locally connected geometric morphisms involves more than just the existence of the extra left adjoint functor, see there.
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.
Every locally connected geometric morphism is a locally cartesian closed functor.
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 above.
If $C$ is a category with all finite limits and if the unique functor $\pi \colon C \to \ast$ to the terminal category is a cover-preserving functor (for $\ast$ equipped with the trivial topology/coverage) then $Sh(C)$ is locally connected. (In particular, this holds for presheaf toposes). 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 .
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
Last revised on May 11, 2022 at 14:18:30. See the history of this page for a list of all contributions to it.