nLab
locally connected topos

Context

Topos Theory

Could not include topos theory - contents

Contents

Idea

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

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

Definition

Definition

An object AA in a topos \mathcal{E} is called a connected object if the hom-functor (A,)\mathcal{E}(A, -) preserves finite coproducts.

Equivalently, an object AA is connected if it is nonempty (noninitial) 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 AA \in \mathcal{E} is a coproduct of connected objects {A i} iI\{A_i\}_{i \in I}, A= iIA iA = \coprod_{i \in I} A_i.

It follows that the index set II is unique up to isomorphism, and we write

π 0(A)=I. \pi_0(A) = I \,.

This construction defines a functor Π 0:ESet:Aπ 0(A)\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

(Π 0LConstΓ):ΓConstΠ 0Set. (\Pi_0 \dashv L Const \dashv \Gamma) : \mathcal{E} \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{Const}{\leftarrow}}{\underset{\Gamma}{\to}}} Set \,.

This is the connected component functor. It generalises the functor, also denoted π 0\pi_0 or Π 0\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 Π 0\Pi_0 already characterizes locally connected toposes.

Proposition

A Grothendieck topos \mathcal{E} is locally connected precisely if the global section geometric morphism Γ:Set\Gamma : \mathcal{E} \to Set is an essential geometric morphism (Π 0LConstΓ):Set(\Pi_0 \dashv L Const \dashv \Gamma) : \mathcal{E} \to Set.

A proof appears as (Johnstone, lemma C.3.3.6).

Proof

Suppose that (Π 0LConstΓ):Set(\Pi_0 \dashv L Const \dashv \Gamma) : \mathcal{E} \to Set exists.

First notice that an object AA is connected in the above sense precisely if Π 0(A)=*\Pi_0(A) = *.

Because for all SSetS \in Set the connectivity condition demands that

(A, SLConst*) S(A,*) S*S \mathcal{E}(A, \coprod_S L Const *) \simeq \coprod_S \mathcal{E}(A,*) \simeq \coprod_S * \simeq S

but by the (Π 0LConst)(\Pi_0 \dashv L Const)-hom-equivalence the first term is

(A,LConst S*)Set(Π 0(A),S) \cdots \simeq \mathcal{E}(A, L Const \coprod_S *) \simeq Set(\Pi_0(A), S)

and the last set is isomorphic to SS precisely for Π 0(A)\Pi_0(A) is the singleton set.

So we need to show that given the extra left adjoint Π 0\Pi_0, every object of \mathcal{E} is a coproduct of objects for which Π 0()\Pi_0(-) is the point.

For that purpose consider for every object AA \in \mathcal{E} the pullback diagram

i A *lim Π 0(A)* lim Π 0(A)* A i A LConstΠ 0(A), \array{ i_A^* {\lim_\to}_{\Pi_0(A)} * &\to& {\lim_\to}_{\Pi_0(A)} * \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ A &\stackrel{i_A}{\to}& L Const \Pi_0 (A) } \,,

where the bottom morphism is the (ΠLConst)(\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

lim sΠ 0(A)(i A ** s)A {\lim_\to}_{s \in \Pi_0(A)} (i_A^* *_s) \stackrel{\simeq}{\to} A

and hence expresses AA as a coproduct of objects i A** si_A * *_s, each of which is a pullback

i A ** s LConst* s A i A LConstΠ 0A, \array{ i_A^* *_s &\to& L Const * \\ \downarrow && \downarrow^{\mathrlap{s}} \\ A &\stackrel{i_A}{\to}& L Const \Pi_0 A } \,,

where the right morphism includes the element ss into the set Π 0A\Pi_0 A. By applying Π 0\Pi_0 to this diagram and pasting on the (Π 0LConst)(\Pi_0 \dashv L Const)-counit we get

Π 0(i A ** s) Π 0LConst* * Π 0(A) Π 0(i A) Π 0LConstΠ 0A Π 0A \array{ \Pi_0(i_A^* *_s) &\to& \Pi_0 L Const * &\to& * \\ \downarrow && \downarrow^{} && \downarrow \\ \Pi_0(A) &\stackrel{\Pi_0(i_A)}{\to}& \Pi_0 L Const \Pi_0 A &\to& \Pi_0 A }

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

Π 0(lim Π 0Ai A ** sA)(lim Π 0AΠ 0(i A ** s)Π 0(A)) \Pi_0( {\lim_{\to}}_{\Pi_0 A} i_A^* *_s \stackrel{\simeq }{\to} A) \simeq ({\lim_\to}_{\Pi_0 A} \Pi_0(i_A^* *_s) \stackrel{\simeq}{\to} \Pi_0(A))

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 Π 0(i A ** s)*\Pi_0(i_A^* *_s) \simeq * for all sΠ 0As \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.

Properties

Characterization over locally connected sites

See at locally connected site.

Equivalent conditions

Definition

For CC and CC cartesian closed categories, a functor F:CDF : C \to D that preserves products is called a cartesian closed functor if the canonical natural transformation

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

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

Proposition

The constant sheaf-functor Δ:𝒮\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 EE is called a connected topos if the left adjoint LConst:SetEL Const : Set \to E is a full and faithful functor.

Proposition

If Γ:ESet\Gamma \colon E\to Set is a locally connected topos, then it is also a connected topos — in that LConstL Const is full and faithful — if and only if the left adjoint Π 0\Pi_0 of LConstL Const preserves the terminal object.

This is (Johnstone, C3.3.3).

Notice that for a connected and locally connected topos, the adjunction

SetΠ 0E Set \stackrel{\overset{\Pi_0}{\leftarrow}}{\hookrightarrow} E

exhibits Set as a reflective subcategory of EE. We may think then of Set as being the localization of EE at those morphisms that induce isomorphisms of connected components.

Examples

Example

For XX a topological space, the category of sheaves Sh(X)Sh(Op(X))Sh(X) \coloneqq Sh(Op(X)) is a locally connected topos precisely if XX is a locally connected space. The functor Π 0\Pi_0 sends a sheaf FSh(X)F \in Sh(X) to the set of connected components of the corresponding etale space.

Example

For C=C = CartSp the site of Cartesian spaces with its good open cover coverage, the topos Sh(CartSp)Sh(CartSp) of smooth spaces is locally connected. An arbitrary XSh(CartSp)X \in Sh(CartSp) is sent to the colimit lim XSet\lim_\to X \in Set. If XX is a diffeological space or even a smooth manifold, then this is the set of connected components of the underlying topological space.

Example

Suppose that CC is a site such that constant presheaves on CC are sheaves. Then the left adjoint Π 0\Pi_0 exists and is given by the colimit functor: if we write L:PSh(C)Sh(C)L : PSh(C) \to Sh(C) for sheafification, then for any sheaf XX, we have

Hom Sh(C)(X,LConstS)Hom PSh(C)(X,LConstS)Hom PSh(C)(X,ConstS)Hom Set(lim X,S). 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 CC is connected, i.e. CC is a locally connected site.

If CC furthermore has a terminal object 11, then the global sections functor Γ:Sh(C)Set\Gamma\colon Sh(C)\to Set (the right adjoint of LConstL Const) is simply given by evaluation at 11, and so the unit SΓLConstSLConstS(1)S \to \Gamma L Const S \cong L Const S(1) is an isomorphism. Thus in this case Sh(C)Sh(C) is additionally connected. This situation also applies to C=CartSpC=CartSp in example 2 above.

Example

If CC is a category with all finite limits and if the unique functor π:C*\pi \colon C \to \ast to the terminal category preserves covers (for *\ast equipped with the trivial topology/coverage) then Sh(C)Sh(C) is locally connected. This is because the inclusion of the terminal object i:*Ci \colon \ast \to C provides a right adjoint to π\pi, so that there is an adjoint quadruple of functors on presheaf categories

(π !Lan π)(π *i !Lan i)(pi *i *)(π !i *Ran i):PSh(C)PSh(*)Sh(C)Set, (\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 ()Lan_{(-)} and Ran ()Ran_{(-)} denote let and right Kan extension, respectively. Now if C*C \to \ast indeed preserves covers and using that C*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)Sh(C) as being locally connected.

But beware that the assumptions here are stronger than they may seem: that C*C \to \ast preserves covers is not automatic, but is a strong condition. It is violated as soon as CC 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)Op(X) of a topological space XX, as in example 1.

References

Section C1.5 and C3.3 of

A variant is in

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)

Revised on June 20, 2013 03:35:39 by Urs Schreiber (82.169.65.155)