nLab locally connected topos

Idea

A topos may be thought of as a generalized topological space. Accordingly, the notions of

• locally connected space

• locally 2-connected space

• etc. …

• locally contractible space

have analogs for toposes and (∞,1)-toposes

• locally connected topos

• locally n-connected (n,1)-topos

• etc. …

Definition

Definition

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.

Definition

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

$$A \;\simeq\; \coprod_{i \in I} A_i \,.$$

If this is the case, it follows that the index set $I$ is unique up to isomorphism, and we denote it by

$$\pi_0(A) \;\coloneqq\; I \,.$$

This construction defines a functor

$$\array{ \Pi_0 \colon & \mathcal{E} &\longrightarrow& Set \\ & A &\mapsto& \pi_0(A) }$$

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.

Proposition

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 $(\Pi_0 \dashv L Const \dashv \Gamma) : \mathcal{E} \to Set$.

Proof

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

(1)$$\Pi_0(A) \;\simeq\; \varnothing \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; LConst \Pi_0(A) \;\simeq\; \varnothing \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; A \xrightarrow{\eta_{A}} \varnothing \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; A \;\simeq\; \varnothing \,,$$

where we used that $LConst$ is a left adjoint and that left adjoints preserve colimits (hence preserve initial objects), we considered the adjunction unit $\eta_A$, and where the last implication follows since the initial object in any topos is strict.

But this gives

$$\Pi_0(A) \simeq \ast \;\;\;\;\;\;\; \Rightarrow \;\;\;\;\;\;\; A \;\text{is connected} \,,$$

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

$$A \;\text{is connected} \;\;\;\;\; \Rightarrow \;\;\;\;\; \Pi_0(A) \simeq \ast$$

observe that the connectivity assumption implies in particular that

(2)$$Set \big( \Pi_0(A) ,\, S \big) \;\simeq\; \mathcal{E} \left( A,\, LConst \coprod_S \ast \right) \;\simeq\; \mathcal{E} \left( A ,\, \coprod_S \, LConst \ast \right) \;\simeq\; {\coprod}_S \, \mathcal{E} \big( A ,\, \ast \big) \;\simeq\; \coprod_S \ast \;\simeq\; S \,,$$

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

$$A \; \text{is connected} \;\;\;\; \Leftrightarrow \;\;\;\; \Pi_0(A) \simeq \ast$$

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

$$\array{ i_A^* {\underset{\underset{\Pi_0(A)}{\longrightarrow}}{\lim}} \ast &\xrightarrow{\phantom{----}}& {\underset{\underset{\Pi_0(A)}{\longrightarrow}}{lim}} \ast \\ {}^{\mathllap{\simeq}}\big\downarrow && \big\downarrow{}^{\mathrlap{\simeq}} \\ A &\xrightarrow{\phantom{--}i_A\phantom{--}}& LConst \Pi_0 (A) \,, }$$

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

$$\underset{ \underset{s \in \Pi_0(A)}{\longrightarrow} }{\lim} (i_A^* \ast_s) \xrightarrow{\phantom{--}\sim\phantom{--}} A$$

and hence expresses $A$ as a coproduct of objects $i_A^* *_s$, each of which is a pullback

$$\array{ i_A^* \ast_s &\xrightarrow{\phantom{----}}& LConst * \\ \big\downarrow && \big\downarrow{}^{\mathrlap{s}} \\ A &\xrightarrow{\phantom{--}i_A\phantom{--}}& LConst \Pi_0 A \,, }$$

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

$$\array{ \Pi_0(i_A^* *_s) &\xrightarrow{\phantom{-----}}& \Pi_0 LConst \ast &\xrightarrow{\phantom{---}}& * \\ \big\downarrow && \big\downarrow^{} && \big\downarrow \\ \Pi_0(A) &\xrightarrow{\phantom{--}\Pi_0(i_A)\phantom{--}}& \Pi_0 LConst \Pi_0 A &\xrightarrow{\phantom{----}}& \Pi_0 A }$$

and by the zig-zag identity the bottom morphism is the identity. This says that in

$$\Pi_0 \left( \underset{\underset{\Pi_0 A}{\longrightarrow}}{\lim} \; i_A^* \ast_s \xrightarrow{\sim} A \right) \simeq \left( \underset{\underset{\Pi_0 A}{\longrightarrow}}{\lim} \; \Pi_0\big(i_A^* \ast_s\big) \xrightarrow{\sim} \Pi_0(A) \right)$$

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 $s \in \Pi_0 A$.

Remark

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.

Properties

Characterization over locally connected sites

See at locally connected site.

Equivalent conditions

Definition

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

$$F(B^A) \to (F(B))^{F(A)}$$

(which is the adjunct of $F(A) \times F(B^A) \simeq F(A \times B^A) \to F(B)$) is an isomorphism.

Proposition

The constant sheaf-functor $\Delta : \mathcal{S} \to \mathcal{E}$ is a cartesian closed functor precisely if $\mathcal{E}$ is a locally connected topos.

Locally connected and connected

A topos $E$ is called a connected topos if the left adjoint $L Const : Set \to E$ is a full and faithful functor.

Proposition

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

$$Set \stackrel{\overset{\Pi_0}{\leftarrow}}{\hookrightarrow} E$$

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.

Examples

Example

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.

Example

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.

Example

Every locally connected geometric morphism is a locally cartesian closed functor.

Example

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

$$Hom_{Sh(C)}(X, L Const S) \simeq Hom_{PSh(C)}(X, L Const S) \simeq Hom_{PSh(C)}(X, Const S) \simeq Hom_{Set}(\lim_\to X, S) \,.$$

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.

Example

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

$$(\pi_! \simeq Lan_\pi) \dashv (\pi^\ast \simeq i_! \simeq Lan_i) \dashv (\pi_\ast \simeq i^\ast ) \dashv (\pi^! \simeq i_* \simeq Ran_i) \;\colon\; PSh(C) \leftrightarrow PSh(\ast) \simeq Sh(C) \simeq Set \,,$$

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 .

Related entries

• essential geometric morphism

• locally connected topos / locally ∞-connected (∞,1)-topos

• connected topos / ∞-connected (∞,1)-topos

• totally connected topos / totally connected (∞,1)-topos

• local topos / local (∞,1)-topos.

• cohesive topos / cohesive (∞,1)-topos

References

Section C1.5 and C3.3 of

• Peter Johnstone, Sketches of an Elephant .

A variant is in

• Marta Bunge, Funk, Quasi locally connected toposes (pdf)

Discussion of characterizations of sites of definition of locally connected toposes is in

• Olivia Caramello, Site characterizations for geometric invariants of toposes, Theory and Applications of Categories, Vol. 26, 2012, No. 25, pp 710-728. (TAC)