nLab locally connected topos

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Topos Theory

topos theory

Background

Toposes

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Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

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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}\big(A, -\big) preserves finite coproducts.

Equivalently, an object AA 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 AA \in \mathcal{E} there exists {A i} iI\big\{A_i \in \mathcal{E}\big\}_{i \in I} such that

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

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

π 0(A)I. \pi_0(A) \;\coloneqq\; I \,.

This construction defines a functor

Π 0: Set A π 0(A) \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 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.

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 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.

Proof

The “only if”-case was just claimed/argued above, we need to show the “if”-case.

Hence suppose that Π 0\Pi_0 with (Π 0LConst)(\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 AA is connected in the above sense precisely if Π 0(A)=*\Pi_0(A) = \ast.

To see this, observe that

(1)Π 0(A)LConstΠ 0(A)Aη AA, \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 LConstLConst is a left adjoint and that left adjoints preserve colimits (hence preserve initial objects), we consideted the adjunction unit η A\eta_A, and where the last implication follows since the initial object in any topos is strict.

But this gives

Π 0(A)*Ais connected, \Pi_0(A) \simeq \ast \;\;\;\;\;\;\; \Rightarrow \;\;\;\;\;\;\; A \;\text{is connected} \,,

because if AA as on the left were the coproduct of non-initial A 1A_1, A 2A_2, then also Π 0(A)\Pi_0(A) would be the coproduct of non-initial Π 0(A 1)\Pi_0(A_1), Π 0(A 2)\Pi_0(A_2), by (1), which would contradict the assumption that Π 0(A)*\Pi_0(A) \simeq \ast.

Conversely, to see

Ais connectedΠ 0(A)* A \;\text{is connected} \;\;\;\;\; \Rightarrow \;\;\;\;\; \Pi_0(A) \simeq \ast

observe that the connectivity assumption implies in particular that

(2)Set(Π 0(A),S)(A,LConst S*)(A, SLConst*) S(A,*) S*S, 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 SSetS \in Set, where in the first step we used the (Π 0LConst)\big(\Pi_0 \dashv LConst\big)-hom isomorphism. But this says that Π 0(A)\Pi_0(A) is connected as an object of Sets, hence that it is the singleton set.

With this equivalence

Ais connectedΠ 0(A)* A \; \text{is connected} \;\;\;\; \Leftrightarrow \;\;\;\; \Pi_0(A) \simeq \ast

in hand (given the extra left adjoint Π 0\Pi_0), we are now reduced to showing that every object of \mathcal{E} is a coproduct of objects for which Π 0()\Pi_0(-) is the point.

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

i A *limΠ 0(A)* limΠ 0(A)* A i A LConstΠ 0(A), \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 (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 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

limsΠ 0(A)(i A ** s)A \underset{ \underset{s \in \Pi_0(A)}{\longrightarrow} }{\lim} (i_A^* \ast_s) \xrightarrow{\phantom{--}\sim\phantom{--}} 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^* \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 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) &\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

Π 0(limΠ 0Ai A ** sA)(limΠ 0AΠ 0(i A ** s)Π 0(A)) \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 Π 0(i A ** s)*\Pi_0(i_A^* *_s) \simeq * for all sΠ 0As \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 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

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

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 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 is a cover-preserving functor (for *\ast equipped with the trivial topology/coverage) then Sh(C)Sh(C) is locally connected. (In particular, this holds for presheaf toposes). 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)(π *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 .

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)

Last revised on May 11, 2022 at 14:18:30. See the history of this page for a list of all contributions to it.