Constructive Complete Distributivity III^{∗}

Robert Rosebrugh Department of Mathematics and Computer Science Mount Allison University Sackville, N. B. Canada

R.J. Wood Department of Mathematics, Statistics and Computing Science Dalhousie University Halifax, N.S. Canada

Dedicated to the memory of our colleague and friend Barry W. Fawcett

Abstract

A complete lattice L is constructively completely distributive, (CCD)(L), if the sup map defined on down closed subobjects has a left adjoint. We characterize preservation of this property by left exact functors between toposes using a “logical comparison transformation”. The characterization is applied to (direct images of) geometric morphisms to show that local homeomorphisms (in particular, product functors) preserve (CCD) objects, while preserving (CCD) objects implies openness.

∗Research partially supported by grants from NSERC Canada. Diagrams typeset using Michael Barr’s diagram macros. AMS Subject Classifications: 18B35, 06D10, 03G10.

Introduction

A complete ordered set L is constructively completely distributive, abbreviated to (CCD)(L), if there is a left adjoint to the sup map ⋁ : DL ⟶ L, where DL is the set of down-closed subobjects of L ordered by inclusion. The condition is equivalent to

(∀S ⊆ DL) (⋀ {⋁ S | S ∈ S} = ⋁ {⋀ {T (S) | S ∈ S} | T ∈ ΠS})

which in turn is the condition (CD)(L) for complete distributivity of L when the power object PL replaces DL. That is, (CCD) is the restriction of (CD) to down-closed subobjects. The name is motivated by the facts [3] that the axiom of choice (ac) implies the equivalence of (CD) with (CCD) for any L, while the equivalence of the two distributivity concepts implies (ac). In short, (CD) ⟺ (CCD) is equivalent to choice. In any topos, PX is (CCD) for all X whereas the statement that PX is (CD) for all X is equivalent to (ac). Hence (CCD) is clearly a relevant notion in most toposes — those where (ac) fails.

The power objects PX are not generally boolean algebras in a topos E and the condition that they are such, denoted (boo)(E), makes a difference in the theory of constructive complete distributivity. We write (CCD^{op})(L) for (CCD)(L^{op}). In [6] we showed that a topos is boolean precisely when the conditions (CCD) and (CCD^{op}) are equivalent for all L.

This article is concerned with studying the preservation of the (CCD) property by left exact functors and geometric morphisms between toposes. Let E be a topos. We denote the 2-category of ordered objects in E by ord(E), where an order is assumed to be reflexive and transitive, but not necessarily antisymmetric, as in [3]. We denote by idl(E) the 2-category with the objects of ord(E), arrows the ideals, and transformations just containments.

Let Γ : E ⟶ S be a left exact functor between toposes. We define a “logical comparison transformation” γ_{E} : ΓD_{E}E ⟶ D_{S}ΓE in ord(S) using ideals and their calculus. This γ_{E} is 2-natural and we show that Γ preserves cocomplete objects in ord(E) precisely if γ_{E} always has a left adjoint. It turns out that Γ preserves (CCD) objects precisely if γ_{E} has a left adjoint which has a left adjoint.

Turning to geometric morphisms, we use the results above to show that direct images of local homeomorphisms preserve (CCD) objects. Geometric morphisms which preserve (CCD) objects are open. Counterexamples show that neither of these implications is reversible, and that essential geometric morphisms need not preserve (CCD) objects. Since local homeomorphisms preserve (CCD) objects, so do arbitrary product functors. Thus a power of a (CCD) objects is a (CCD) object. Furthermore, the cotensor ord(E)(X, L), for L a (CCD) object and X in ord(E), is a (CCD) object [6].

1 Left Exact Functors

Let E and S be elementary toposes and Γ : E ⟶ S a left exact functor. It induces a 2-functor ord(Γ) : ord(E) ⟶ ord(S) which we abbreviate to Γ. It is also convenient to define the 2-category idl(E) whose objects are those of ord(E), whose arrows are (order) ideals and whose transformations are “containments”. For ordered objects X and Y, an ideal from X to Y is a subobject r : R[figure] Y × X which is “down-closed in Y ” and “up-closed in X”. In terms of “elements”, we mean that if y′ ≤ y and yRx and x ≤ x′, then y′ ≤ x′. We write r ≤ s to indicate that r factors through s : S[figure] Y × X. Horizontal composition is given by composition of relations.

While ord(E) is locally ordered, idl(E) is locally antisymmetrically ordered as a result of our having defined ideals in terms of subobjects rather than just monomorphisms. A word about our apparent switch in generality may be helpful. Given an ordered object X in E and x, y in X we write x ≅ y if and only if both x ≤ y and y ≤ x. The associated antisymmetric ordered object is X/≅ and we have q : X ⟶ X/≅ in ord(E). However, we cannot conclude that q is an equivalence of ordered objects because there may be no splitting of q in E. Now, for any object E in E write mono(E) for the ordered set of monomorphisms with codomain E and sub(E) for the antisymmetric ordered set of subobjects of E. Then mono(E) ⟶ sub(E) is an instance of q in a suitably large category of sets. We do not assume choice for the latter, but if we regard the pullbacks of “true”, τ : 1 ⟶ Ω, in E as being canonically specified (which we regard as a reasonable working convention), then mono(E) ⟶ sub(E) has a canonical splitting for each E and it seems sensible to use it. Were we to carry out this discussion in an arbitrary regular category we would adopt the approach of [2].

Ideals X ⟶ Y are in order-isomorphic correspondence with ord(E) arrows Y^{op} ×X ⟶ Ω which in turn are in order isomorphic correspondence with ord(E) arrows X ⟶ DY. We write

( )^{∧} : idl(E)(X, Y) [figure: ∼] ord(E)(X, DY) : ( )^{∨}

and note that y ∈ φ^{∧}x if and only if yφx; yh^{∨}x iff y ∈ hx. If f : X ⟶ Y in ord(E) we define f_{+} : X ⟶ Y in idl(E) by yf_{+}x if and only if y ≤ fx. Similarly, f^{+} : Y ⟶ X is defined by xf^{+}y if and only if fx ≤ y. We have f_{+} ⊣ f^{+} in idl(E) and ( )_{+} : ord(E) ⟶ idl(E) is proarrow equipment. (See [7, 8].) Sometimes we suppress ( )_{+} when we feel that it does not cause confusion. Note that applying Γ to subobjects does not give a 2-functor idl(Γ) : idl(E) ⟶ idl(S) unless Γ happens to preserve images. However, (together with ord(Γ)) it does give an arrow of “F” as described in great detail in [1]. Abbreviating idl(Γ) by Γ, we record those facts about idl(Γ) which we need for this paper:

(i) Γ(1_{X}) = 1_{ΓX} for all X in idl(E) (ii) Γ(f_{+}) = (Γf)_{+} for all f in ord(E) (iii) Γ(f^{+}) = (Γf)^{+} for all f in ord(E) (iv) Γ(φf) = ΓφΓf for all f in ord(E) and φ in idl(E)

(v) Γ(f^{+}φ) = Γf^{+}Γφ for all f in ord(E) and φ in idl(E)

Note too that ord(Γ) preserves full faithfulness of an arrow f : X ⟶ Y in ord(E). For f is fully faithful if and only if f^{+}f = 1_{X}.

As usual, we write ↓_{E} : E ⟶ D_{E}E in ord(E) for the down-segment (yoneda) arrow given by ↓_{E}(x) = ↓x = {y ∈ E | y ≤ x}. We define γ_{E} : ΓD_{E}E ⟶ D_{S}ΓE in ord(S) by γ_{E} = (Γ↓_{E})^{+}^{∧}. Lemma 1 ( )^{∨} : ord(E)(X, DY) ⟶ idl(E)(X, Y) is given by composition with the arrow ↓_{Y}^{+}: DY ⟶ Y. That is

(X ⟶^{h} DY)^{∨} = X ⟶^{h_{+}} DY ⟶^{↓_{Y}^{+}} Y

Proof. We have

y(↓_{Y}^{+}h_{+})x iff (∃T ∈ DY)(y↓_{Y}^{+}T h_{+}x)
iff (∃T ∈ DY)(↓_{Y} (y) ⊆ T ⊆ hx)
iff  (↓_{Y} (y) ⊆ hx)
iff  (y ∈ hx)

Corollary 2 For X ⟶^{f} Y and Y ⟶^{h} DZ in ord(E), (hf)^{∨} = h^{∨}f_{+} in idl(E).

Proof.

(hf)^{∨} = ↓_{Z}^{+}(hf)_{+} = ↓_{Z}^{+}h_{+}f_{+} = h^{∨}f_{+}

Lemma 3 For φ : F ⟶ E in idl(E)

[figure: ΓF; D_{S}ΓE; (Γφ)^{∧}; ΓD_{E}E; Γφ^{∧}; γ_{E}]

Proof. It suffices to show Γφ = (γ_{E} · Γφ^{∧})^{∨}

(γ_{E} · Γφ^{∧})^{∨} = ((Γ↓_{E}^{+})^{∧} · Γφ^{∧})^{∨}
= Γ↓_{E}^{+} · Γφ^{∧}  (by Corollary 2)
= Γ(↓_{E}^{+} · φ^{∧})
= Γ(φ^{∧}^{∨})  (by Lemma 1)
= Γφ

Corollary 4 For E in E

[figure: ΓE; D_{S}ΓE; ↓_{ΓE}; ΓD_{E}E; Γ↓_{E}; γ_{E}]

in ord(S).

Proof. (E ⟶^{1_{E}} E)^{∧} = ↓_{E} : E ⟶ D_{E}E.

Lemma 5 For h : X ⟶ DY and g : Z ⟶ Y in ord(E), (Dg · h)^{∨} = g^{+}h^{∨} in idl(E). Proof. Consider

[figure: X; h; Y; Z; g; DY; DZ; g_{!}; ↓_{Y}; ↓_{Z}]

where g_{!} denotes the left adjoint to Dg : DY ⟶ DZ. (D is given on arrows by inverse image.) The square commutes because ↓ is natural from the identity to ( )_{!}. Taking right adjoints of the arrows in the square, the result follows immediately from Lemma 1.

Corollary 6 For E in E

[figure: ΓD_{E}E; D_{S}ΓE; γ_{E}; D_{S}ΓD_{E}E; ↓_{ΓDEE}; D_{S}Γ↓_{E}]

Proof.

(D_{S}Γ↓_{E} · ↓_{ΓDEE})^{∨} = Γ↓_{E}^{+} · ↓_{ΓDEE}^{∨} by Lemma 5
= Γ↓_{E}^{+} · 1_{ΓDEE}
= γ_{E}^{∨}

Lemma 7 For Γ : E ⟶ S left exact, γ is a 2-natural transformation:

[figure: ord(S)^{coop}; ord(S); D_{S}; ord(E)^{coop}; ord(E); D_{E}; ord(Γ)^{coop}; ord(Γ); γ]

Proof. Because ord(S) is a locally ordered 2-category, it suffices to prove ordinary naturality of γ. Let f : X ⟶ Y be an arrow of ord(E) and consider:

[figure: D_{S}ΓY; D_{S}ΓX; D_{S}Γf; ΓD_{E}Y; ΓD_{E}X; ΓD_{E}f; γ_{Y}; γ_{X}]

(γ_{X} · ΓD_{E}f)^{∨} = ((Γ↓_{X})^{+}^{∧} · ΓD_{E}f)^{∨}
= ((Γ↓_{X})^{+}(ΓD_{E}f)_{+} (Lemma 2)
= Γ(↓_{X}^{+} D_{E}f)
= Γ(f^{+}· ↓_{Y}^{+})
= Γf^{+} · (Γ ↓_{Y})^{+}
= ((D_{S}Γf)(Γ ↓_{Y})^{+}^{∧})^{∨} (Lemma 5)
= ((D_{S}Γf)γ_{Y})^{∨}.

Theorem 8 For Γ : E ⟶ S a left exact functor between toposes, the following statements for all E in ord(E) are equivalent:

(i) γ_{E} has a left adjoint

(ii) E cocomplete ⇒ γ_{E} has a left adjoint

(iii) E cocomplete ⇒ ΓE cocomplete

(iv) ΓD_{E}E is cocomplete.

Proof. (i) ⇒ (ii) Trivial. (ii) ⇒ (iii) E is cocomplete iff ↓_{E} : E ⟶ D_{E}E has a left adjoint (⋁_{E}). The implication follows immediately from Lemma 4 since ord(Γ) preserves adjunctions. (iii) ⇒ (iv) Trivial, since D_{E}E is cocomplete for all E in ord(E). (iv) ⇒ (i) Consider the factorization of γ_{E} given in Corollary 6. The left adjoint of D_{S}Γ↓_{E} is, in any event, (Γ↓_{E})_{!}. Cocompleteness of ΓD_{E}E means, precisely, the existence of a left adjoint for ↓_{ΓDEE} .

Our goal is change of base criteria for (CCD) objects. Write pamord(E) for the locally full sub-2-category of ord(E) determined by the complete objects of ord(E) and the infpreserving arrows between them and define the cocomplete objects therein to be those E for which ↓_{E} : E ⟶ D_{E}E has a left adjoint in pamord(E). Since an arrow is inf-preserving precisely if it has a left adjoint we see that (CCD) objects of ord(E) are the cocomplete objects of pamord(E) and this suggests that we structure our preservation of (CCD) criteria parallel to those of Theorem 8.

Such an approach also suggests the question of a left exact version. For if we write lexord(E) for the locally full sub-2-category of ord(E) determined by the finitely complete objects of ord(E) and the left exact arrows between them and define cocompleteness of E in lexord(E) to mean the existence of a left adjoint to ↓_{E} in lexord(E), then the cocomplete objects of lexord(E) are the locales of ord(E). We write (LOC)(E) for the statement “E is a locale” and record without proof

Theorem 9 For Γ : E ⟶ S a left exact functor between toposes the following steatements for all E in lexord(E) are equivalent:

(i) γ_{E} has a left exact left adjoint

(ii) (LOC)(E) ⇒ γ_{E} has a left exact left adjoint

(iii) (LOC)(E) ⇒ (LOC)(ΓE)

(iv) (LOC)(ΓD_{E}E).

In fact we can say somewhat more.

Theorem 10 For Γ : E ⟶ S a left exact functor between toposes, each of the equivalent statements of Theorem 8 is equivalent to each statement of Theorem 9.

Proof. ((iii) of Theorem 8 ⇒ (iii) of Theorem 9) Let E be finitely complete. If E is a locale then certainly by (iii) of Theorem 8, ΓE is cocomplete. However, Γ, being left exact, preserves Heyting algebras; so if E is a locale then ΓE is a locale. ((iv) of Theorem 9 ⇒ (iv) of Theorem 8). Let E be an object of ord(E). We have ↓_{E} : E[figure] D_{E}E, fully faithful, and hence, via Γ(↓_{E})_{!} ⊣ ΓD_{E}↓_{E}, ΓD_{E}E is coreflective in ΓD_{E}D_{E}E. For any E, D_{E}E is in lexord(E); by assumption ΓD_{E}D_{E}E is a locale and hence cocomplete; finally ΓD_{E}E is complete and hence cocomplete.

Theorem 11 For Γ : E ⟶ S a left exact functor between toposes the following statements for all E in pamord(E) are equivalent:

(i) γ_{E} has a left adjoint which has a left adjoint.

(ii) (CCD)(E) ⇒ γ_{E} has a left adjoint which has a left adjoint.

(iii) (CCD)(E) ⇒ (CCD)(ΓE)

(iv) (CCD)(ΓD_{E}E).

Proof. (i) ⇒ (ii) Trivial. (ii) ⇒ (iii) (CCD)(E) iff ↓_{E} has a left adjoint which has a left adjoint. The implication follows immediately from Corollary 4 (iii) ⇒ (iv) Trivial, since (CCD)(D_{E}E) for all E in ord(E). (iv) ⇒ (i) Consider the factorization of γ_{E} given in Corollary 6. Here we have

(Γ⋁_{E})_{!} ⊣ (Γ↓_{E})_{!} ⊣ D_{E}Γ↓_{E} and ⇓_{ΓDEE} ⊣ ⋁_{ΓDEE} ⊣ ↓_{ΓDEE}

where, as in [3], we write ⇓⊣ ⋁ for (CCD) objects.

Remark One might enquire whether condition (i) in each of Theorems 8, 9 and 11 is expressible in terms of the power object functor P. For Theorem 8 this is easy. For any X in E we have PX = DDX, where D : E ⟶ ord(E) is the discrete order functor, left adjoint to the forgetful functor, | |. Certainly, then (i) of Theorem 8 implies

(∀X in E)(γ_{X} : ΓP_{E}X ⟶ P_{S}ΓX has a left adjoint)

(where we have abbreviated γ_{DX} by γ_{X} and noted that ord(Γ) preserves discreteness.) This condition implies (i) of Theorem 8. For any E in ord(E) we have the counit for the adjunction, D ⊣ | |, D|E| ⟶ E. Invoking the left adjoint to D of this arrow, which is down closure, ( )^{}, and naturality of γ, Lemma 7, it is easy to describe a left adjoint for γ_{E} in terms of a left adjoint for γ_{D|E|}.

The situation in Theorems 9 and 11 is not so simple with respect to γ, however the following characterization is also useful.

Theorem 12 For Γ : E ⟶ S a left exact functor between toposes,

(∀E in ord(E))((CCD)(E) ⟹ (CCD)(ΓE))

if and only if (∀X in E)((CCD)(ΓPX)).

Proof. Power objects are always (CCD) . For the converse, invoke the Raney-Buchi presentation of (CCD) objects:

[figure: E; DE; ⊥; ⊥; P(D|E|); ⊥; ⊥]

(see [3]) and note that ord(Γ) preserves adjunction and full faithfulness. Then (CCD)(ΓE) follows from Propositions 10 and 11 of [3].

It is clear that Γ preserves D (in the sense that γ is an isomorphism) if and only if Γ preserves P. Hence:

Theorem 13 Logical functors between toposes preserve (CCD) objects (and, of course, locales and complete objects.)

2 Geometric Morphisms

We turn now to a brief discussion of geometric morphisms.

Let Γ : E ⟶ S be geometric with inverse image Δ : S ⟶ E, unit η : 1_{S} ⟶ ΓΔ and counit ε : ΔΓ ⟶ 1_{E}. The previous considerations for Γ apply to Δ too and we write δ_{S} : ΔD_{S}S ⟶ D_{E}ΔS for (Δ↓_{S})^{+}^{∧}. We refer to γ, respectively δ, as the logical comparison transformation associated to Γ, respectively Δ.

Lemma 14 For Γ : E ⟶ S geometric,

[figure: ΓD_{E}E; ΓD_{E}ΔΓE; ΓD_{E}ε_{E}; ΓΔD_{S}ΓE; Γ(δ_{ΓE}^{+}); D_{S}ΓE; (η_{DSΓE})^{+}; ^{γ_{E}^{∨}}; ΓE; ↓_{ΓE}^{+}]

Proof. Consider the following diagram:

[figure: 1_{ΓE}; ΓD_{E}E; ΓD_{E}ΔΓE; ΓD_{E}ε_{E}; ΓΔD_{S}ΓE; Γ(δ_{ΓE}^{+}); D_{S}ΓE; (η_{DSΓE})^{+}; ΓΔΓE; Γ↓_{ΔΓE}^{+}; ΓE; Γ↓_{E}^{+}; ΓΔΓE; ΓΔ↓_{ΓE}^{+}; ΓE; ↓_{ΓE}^{+}; Γε_{E}^{+}; (η_{ΓE})^{+}]

The right hand square commutes by naturality of η and application of ( )^{+}. The middle triangle is an instance of Lemma 4 (followed by application of ( )^{+} and Γ). For the left hand trapezoid recall that ↓ is a natural transformation from the identity to ( )_{!} and consider the ε_{E}’th instance of the naturality square; take right adjoints, noting that (ε_{E})_{!}^{+} = D_{E}ε_{E}; apply Γ. The bottom region of the diagram commutes by application of one of the triangular identities for the Δ ⊣ Γ adjunction. Finally,

γ_{E}^{∨} = Γ↓_{E}^{+}^{∧}^{∨} = Γ↓_{E}^{+}.

Corollary 15 If

[figure: ΓD_{E}ΔΓE; ΓΔD_{S}ΓE; Γ(δ_{ΓE}^{+}); D_{S}ΓE; (η_{DSΓE})^{+}]

is ( ( )_{+} of ) an arrow of ord(S) then

γ_{E} = (η_{DSΓE})^{+} · Γ(δ_{ΓE})^{+} · ΓD_{E}ε_{E}.

Proof. Given the hypothesis, the composite of the three arrows shown horizontally in the statement of Lemma 14 is ( ( )_{+} of ) an arrow of ord(S). By Lemma 1, ( )^{∨} applied to it is given by postcomposition with ↓_{ΓE}^{+}. The result follows since ( )^{∨} is an order isomorphism.

Theorem 16 Direct images of local homeomorphisms preserve (CCD) objects.

Proof. Let Γ : E ⟶ S be a local homeomorphism (that is, assume Δ is logical.) In this case δ is an isomorphism, so (δ_{ΓE})^{+} = δ_{ΓE}^{−1}. Also in this case, by [5, Sublemma p. 88] both η and ε have both left and right adjoints at all complete arguments. D_{S}ΓE is complete so (η_{DSΓE})^{+} is an arrow of ord(S) and the hypothesis of Corollary 15 is satisfied so that γ_{E} = (η_{DSΓE})^{+} · Γ(δ_{ΓE})^{+} · ΓD_{E}ε_{E}. It suffices to show that each of the above three factors have left adjoints which have left adjoints for cocomplete E. The middle factor, being an isomorphism, trivially does. By the result of [5] mentioned above, (η_{DSΓE})^{+} satisfies the condition for all E. For all E we have Γ(ε_{E})_{!} ⊣ ΓD_{E}ε_{E}. For (co)complete E we have λ_{E} ⊣ ε_{E}, again by [5] and, in this case, Γ(λ_{E})_{!} ⊣ Γ(ε_{E})_{!} too.

Corollary 17 For all α : I ⟶ J in S, the functor Π_{α} : S/I ⟶ S/J preserves (CCD) objects. In particular, Π_{I} : S/I ⟶ S and − × − : S × S ⟶ S preserve (CCD) objects.

Corollary 18 If L is a (CCD) object of ord(S) and I is in S then L^{I} is a (CCD) object of ord(S).

Corollary 19 If L is a (CCD) object of ord(S) and X is in ord(S) then ord(S)(X, L) is a (CCD) object of ord(S).

Proof. The canonical arrow D|X| ⟶ X in ord(S) allows us to define another arrow ord(S)(X, L) ⟶ ord(S)(D|X|, L) ≅ L^{|X|}, the inclusion of order preserving arrows in all arrows from X to L. Since L is complete the latter has both left and right adjoints, given by Kan extensions. Since L^{|X|} is (CCD) the adjoint string exhibits ord(S)(X, L) as a (CCD) object. See [3, Proposition 10].

It is not true that direct images of essential geometric morphisms necessarily preserve (CCD) objects.

Counterexample 1 Consider Φ : set ⟶ set^{2} given by ΦX = (X ⟶ 1). We have Δ ⊣ Γ ⊣ Φ where Γ(X : X_{0} ⟶ X_{1}) = X_{0} and ΔX = (X ⟶^{=} X). Abbreviate D_{set2} by D_{2} and D_{set} by D. Note that D_{2}(X : X_{0} ⟶ X_{1}) is given by (D_{2})_{0} ⟶ DX_{1}, where (D_{2})_{0} is the set of all S_{0} ⟶ S_{1} with S_{0} in DX_{0}, S_{1} in DX_{1} satisfying

[figure: S_{1}; X_{1}; S_{0}; X_{0}]

and D_{2}X(S_{0} ⟶ S_{1}) = S_{1}. In particular, (D_{2}ΦX) = D_{2}(X ⟶ 1) can be regarded as 1 + DX ⟶ 2 = {0 < 1} with ⊥ ∈ 1 and ⊥ < S for all S in DX; ⊥ ↦ 0; S ↦ 1. Writing φ for the logical comparison transformation, φ_{X} : ΦDX ⟶ D_{2}ΦX can be shown to be

[figure: 1; 2; 1; DX; 1 + DX; i]

where i is the injection into the sum. This i has a left adjoint (⊥ ↦ ∅, S ↦ S) which has a left adjoint (∅ ↦ ⊥, ∅ ≠ S ↦ S). However, the latter does not commute with the left adjoint to the left adjoint of 1 ( which is 0 ) for non-empty X. Thus φ_{X} does not have a left adjoint in ord(set^{2}) for any complete X in ord(set).

We also note that the condition “γ_{1} has a left adjoint which has a left adjoint”, for Γ geometric, is precisely openness as defined in [4, p. 56]. Thus geometric morphisms which preserve (CCD) objects are open. However, neither the converse of this remark nor the converse of Theorem 16 is valid.

Counterexample 2 Let L be a locale in set which is not (CCD) . For example, take L to be the lattice of open subsets of the reals. Then the global sections functor Γ : sh(L) ⟶ set is open, as is any geometric morphism with codomain set, but Γ(Ω_{sh(L)}) ≅ L implies that Γ does not preserve (CCD) objects.

Counterexample 3 Consider the global sections (or domain) functor Γ : set^{2} ⟶ set. It is evidently not a local homeomorphism. We show that it does preserve (CCD) objects. Let X : X_{0} ⟶ X_{1} be any object of set^{2}. It is well known that P_{set2}X is the projection shown in the following lax limit diagram in ord(set).

[figure: γ_{DX}; ΓP_{set2}X; P_{set}X_{1}; P_{set2}X; P_{set}X_{0}; ∃_{X}; = P_{set}ΓX; ≥]

(Here ∃_{X} is direct image, the left adjoint of P_{set}X = X^{−1}. A routine calculation shows that the other projection is a component of the logical comparison transformation as suggested by the above notation.)

Thus ΓP_{set2}X = {(S_{0}, S_{1}) ∈ PX_{0} × PX_{1}|X(S_{0}) ⊆ S_{1}}. It is easy to show that the projections paired give ΓP_{set2}X[figure] PX_{0} × PX_{1} with left adjoint given by the formula: (T_{0}, T_{1}) ↦ (T_{0}, X(T_{0}) ⋃(T_{1}) and right adjoint given by: (T_{0}, T_{1}) ↦ (T_{0} ⋂ X^{−1}(T_{1}), T_{1}). By Corollary 17, PX_{0} × PX_{1} is (CCD) . By [3, Proposition 10], ΓP_{set2}X is (CCD) . By Theorem 11, Γ preserves (CCD) objects.

This last Counterexample together with Counterexample 1 also show that a right adjoint of a (CCD) preserving functor, even if the latter is geometric, need not be (CCD) preserving. Indeed, domain : set^{2} ⟶ set has right adjoint the Φ of Counterexample 1.

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