Sheaves of Structures, Heyting-Valued Structures,and a Generalization of Łos’s Theorem

Hisashi Aratake*

Research Institute for Mathematical Sciences,Kyoto University, Kyoto, Japan

AbstractSheaves of structures are useful to give constructions in universal al-

gebra and model theory. We can describe their logical behavior in termsof Heyting-valued structures. In this paper, we first provide a systematictreatment of sheaves of structures and Heyting-valued structures fromthe viewpoint of categorical logic. We then prove a form of Łos’s the-orem for Heyting-valued structures. We also give a characterization ofHeyting-valued structures for which Łos’s theorem holds with respect toany maximal filter.

0 Introduction

Sheaf-theoretic constructions have been used in universal algebra and modeltheory. In this context, sheaves of abelian groups or rings in geometry aregeneralized to sheaves of structures. We can obtain, for example, the product(resp. an ultraproduct) of a family of structures from some sheaf by takingthe set of global sections (resp. a stalk). This viewpoint originated from theearly literature [Com74], [Ell74] and [Maci73]. In combination with the theoryof sheaf representations of algebras, Macintyre [Maci73] succeeded in giv-ing model-companions of some theories of commutative rings by transferringmodel-theoretic properties from stalks to global sections.

On the other hand, sheaves have another description as Heyting-valuedsets. The notion of Heyting-valued sets originally arises from that of Boolean-valued models of set theory, which was introduced in relation to Cohen’sforcing. The development of topos theory in the early seventies, mainly dueto Lawvere & Tierney, revealed profound relationships between toposes andmodels of set theory; objects in a topos can be regarded as “generalized sets”in a universe. Subsequently, Fourman & Scott [FS79] and Higgs 1 indepen-dently established the categorical treatment of Heyting-valued sets (see Re-mark 1.6). The category Sh(X) of sheaves of sets on a space X and the cate-gory Set(O(X)) ofO(X)-valued sets turned out to be categorically equivalent.

*E-mail address: aratake@kurims.kyoto-u.ac.jp

1Originally in his preprint written in 1973, part of which was later published as [Hig84].

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Some model-theorists of that era immediately applied Heyting-valued setsto concrete problems in sheaf-theoretic model theory. However, general meth-ods of Heyting-valued model theory have not been explored enough, thoughFourman & Scott mentioned such a direction in the preamble of [FS79]. Evenworse, we are not aware of any clear explanation of the relationship betweensheaves of structures and Heyting-valued structures. In this paper, employ-ing well-established languages of categorical logic, we will give a gentle andcoherent account of Heyting-valued semantics of first-order logic from thecategorical point of view, and will apply that framework to obtain a gener-alization of Łos’s theorem for Heyting-valued structures. We also provide acharacterization of Heyting-valued structures for which Łos’s theorem holdsw.r.t. any maximal filter. Our theorems improve the works of Caicedo [Cai95]and Pierobon & Viale [PV20]. While our principal examples of Heyting-valuedsets are sheaves on topological spaces, other natural examples include sheaveson the complete Boolean algebra of regular open sets and Boolean-valued setson the measure algebra. Therefore, we will develop our theory based on anycomplete Heyting algebra (a.k.a. a frame or a locale) rather than a topologicalspace.

The intended audience for this paper is anyone who has interests bothin model theory and in categorical logic. We assume some familiarity withtopos theory and first-order categorical logic. Most categorical prerequisitesare covered by [SGL]. In §2.1, we will recall some elements of first-ordercategorical logic.

The areas related to this paper (and its sequels in the future) are diverse,including model theory, universal algebra, set theory, categorical logic, topostheory, and their applications to ordinary mathematics. The author did hisbest to ensure that the reader can follow the scattered literature (especially inmodel theory and topos theory) on each occasion during the course.

The structure of this paper: In §1, we will begin with preliminaries onsheaves and Heyting-valued sets. After we see basic properties of Heyting-valued sets and morphisms between them, we will give an outline of theequivalence of sheaves and Heyting-valued sets. We also provide some de-tails on the topos Set(O(X)) of O(X)-valued sets. In §2, we will study struc-tures in the toposes Sh(X) and Set(O(X)) and the relationship between them.We will also introduce forcing values of formulas categorically. In §3, ob-serving that sheaves of structures generalize some model-theoretic construc-tions, we will introduce a further generalization of filter-quotients of sheavesto Heyting-valued structures and prove Łos’s theorem and the characteriza-tion theorem. In §4, we will indicate some possible future directions with anexpanded list of previous works.

Closing the introduction, we have to mention Loullis’ work [Lou79] onBoolean-valued model theory. The starting point of this research was try-ing to digest his work from a modern categorical viewpoint, though our workis still too immature to give the reader a full explanation of his contribution.If the author had not met his work, this paper would not have existed. The

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author regrets his untimely death, according to [BR81], in 1978.

Acknowledgment: The author is grateful to his supervisor, Kazushige Terui,for careful reading of an earlier draft and many helpful suggestions on thepresentation of this paper. He also thanks Soichiro Fujii for useful comments.

1 Sheaves and Heyting-Valued Sets

Heyting-valued sets were introduced independently by Higgs [Hig84] andby Fourman & Scott [FS79]. In this section, we will review the constructionof the category Set(O(X)) of O(X)-valued sets for a locale X , its relationto sheaves on X , and its categorical structures as a topos. Most results arecovered by [Hig84], [FS79], [Elephant, §C1.3] and [HoCA3, Chapter 2]. Forthe reader’s convenience, we will occasionally give brief sketches of proofs.

For aspects of Heyting-valued sets in intuitionistic logic, see [TvD88, Chap-ters 13–14]. As a category, Set(O(X)) is a prototypical example of the toposobtained from a tripos (see [HJP80] and [vOos08, Chapter 2]). The internallogic of Set(O(X)) is reduced to the logic of tripos. Walters [Wal81], [Wal82]developed another direction of generalization of Heyting-valued sets.

1.1 Heyting-Valued Sets

Definition 1.1. A frame is a complete lattice satisfying the infinitary distribu-tive law:

a ∧∨i

bi =∨i

a ∧ bi.

In particular, any frame has 0 and 1.

A frame is the same thing as a complete Heyting algebra: the infinitarydistributive law for a frame H says that each monotone map a ∧ (−) : H → H

preserves arbitrary joins. This happens exactly when each map a ∧ (−) has aright adjoint a→ (−) : H → H , i.e., a monotone map satisfying

∀b, c ∈ H, [a ∧ b ≤ c ⇐⇒ b ≤ a→ c].

This fact follows either from category theory (the General Adjoint Functor The-orem), or from a direct construction

a→ c :=∨ b ∈ H ; a ∧ b ≤ c .

On the other hand, frame homomorphisms differ from those for completeHeyting algebras (and even those for complete lattices):

Definition 1.2. Let H,H ′ be frames. A frame homomorphism h : H → H ′

is a map from H to H ′ preserving finite meets and arbitrary joins. Let Frmdenote the category of frames.

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Similarly to the above, any frame homomorphism h : H → H ′ has a rightadjoint k : H ′ → H given by

k(b) =∨ a ∈ H ; h(a) ≤ b .

Any continuous map f : X → Y of topological spaces gives rise to aframe homomorphism f∗ : O(Y ) → O(X) given by f∗(V ) = f−1(V ), whereO(X) (resp. O(Y )) is the frame of open sets of X (resp. Y ). The functorO(−) : Topop → Frm is full and faithful on sober spaces (i.e., spaces satisfy-ing a suitable axiom between T0 and T2). Therefore, it translates the languageof spaces to that of frames, and we may consider frames as “point-free” spaces.This justifies the following definition:

Definition 1.3. A frame considered as an object of Frmop is called a locale.We denote Frmop by Loc and the frame corresponding to a locale X ∈ Loc

by O(X). We will write U, V, etc. for elements of O(X) and 0X (resp. 1X ) forthe smallest (resp. largest) element.

For a morphism f : X → Y of locales, the corresponding frame homomor-phism is denoted by f∗ : O(Y ) → O(X). f∗ has a right adjoint f∗ : O(X) →O(Y ). Morphisms of locales are also called continuous maps of locales.

By writing X` for the locale given by a topological space X , i.e., O(X`) =

O(X), we now have a functor (−)` : Top → Loc. It has a right adjointpt: Loc → Top sending a locale X to the space pt(X) of “points of X”(see, e.g., [SGL, Chapter IX]). For more on frames and locales in point-freetopology, see [Joh82] and [PP12].

We are now ready to define Heyting-valued sets. In the remainder of thissection, we fix a locale X .

Definition 1.4. An O(X)-valued set (A,α) is a pair of a set A and a mapα : A×A→ O(X) such that

• ∀a, b ∈ A, α(a, b) = α(b, a)

• ∀a, b, c ∈ A, α(a, b) ∧ α(b, c) ≤ α(a, c)

In the logic of tripos (associated with X), α is a “partial equivalence rela-tion” on A. Instead of α(a, b), the notation Ja = bK is frequently used in theliterature. We introduce a few conventions:

• α(a) := α(a, a) is called the extent of a. Note that α(a, b) ≤ α(a) ∧ α(b).

• If α(a) = 1X , then a is called a global element of (A,α).

Morphisms of Heyting-valued sets should be “functional relations” (againin the logic of tripos).

Definition 1.5. Let (A,α), (B, β) beO(X)-valued sets. A morphism ϕ : (A,α)→(B, β) of O(X)-valued sets is a map A×B → O(X) which satisfies

∀a, a′ ∈ A, ∀b, b′ ∈ B, α(a, a′) ∧ ϕ(a, b) ∧ β(b, b′) ≤ ϕ(a′, b′),

∀a ∈ A, ∀b, b′ ∈ B, ϕ(a, b) ∧ ϕ(a, b′) ≤ β(b, b′),

∀a ∈ A, α(a) =∨b∈B

ϕ(a, b).

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In particular, ϕ(a, b) ≤ α(a) ∧ β(b) always holds. If ψ : (B, β) → (C, γ) isanother morphism, we can define the composite ψ ϕ by

(ψ ϕ)(a, c) =∨b∈B

ϕ(a, b) ∧ ψ(b, c).

We write Set(O(X)) for the category of O(X)-valued sets and morphisms,where the identity id(A,α) is given by α itself.

Remark 1.6. In set theory, for a frame H , we can construct a model V (H) ofintuitionistic set theory ( [Bel14, Chapter IV]). V (H) is called the Heyting-valued universe. The category Set(H) is regarded as a categorical counter-part of V (H) ( [Bel05, Appendix], [ACM19]), and we can take arguments andexamples from set theory to investigate Heyting-valued sets (cf. [PV20]).

We list useful facts on Heyting-valued sets, some of which will not be usedin this paper.

Lemma 1.7. Two morphisms ϕ,ψ : (A,α)⇒ (B, β) are identical if

∀a ∈ A, ∀b ∈ B, ϕ(a, b) ≤ ψ(a, b).

Proof. Suppose ∀a ∈ A, ∀b ∈ B, ϕ(a, b) ≤ ψ(a, b). Then,

ψ(a, b) = ψ(a, b) ∧ α(a) = ψ(a, b) ∧∨b′

ϕ(a, b′)

≤∨b′

[ϕ(a, b′) ∧ ψ(a, b′) ∧ ψ(a, b)] ≤∨b′

[ϕ(a, b′) ∧ β(b′, b)] = ϕ(a, b).

Proposition 1.8. Let ϕ : (A,α)→ (B, β) be a morphism in Set(O(X)).

(1) ϕ is a monomorphism if and only if

∀a, a′ ∈ A, ∀b ∈ B, ϕ(a, b) ∧ ϕ(a′, b) ≤ α(a, a′).

(2) ϕ is an epimorphism if and only if

∀b ∈ B, β(b) =∨a∈A

ϕ(a, b).

(3) ϕ is an isomorphism if and only if it is monic and epic. In other words,Set(O(X)) is a balanced category. If ϕ is an isomorphism, ϕ−1 is givenby ϕ−1(b, a) = ϕ(a, b).

Definition 1.9. We say that a morphism ϕ : (A,α)→ (B, β) is represented bya map h : A→ B when

∀a ∈ A, ∀b ∈ B, ϕ(a, b) = α(a) ∧ β(ha, b).

Proposition 1.10.

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(1) A morphism ϕ : (A,α) → (B, β) is represented by a map h : A → B ifand only if

∀a ∈ A, ∀b ∈ B, ϕ(a, b) ≤ β(ha, b).

(2) A map h : A → B represents some morphism from (A,α) to (B, β) ifand only if

∀a, a′ ∈ A, α(a, a′) ≤ β(ha, ha′).

Moreover, if h further satisfies α(a) = β(ha) for all a ∈ A, then themorphism ϕ represented by h is given simply by ϕ(a, b) = β(ha, b).

(3) Suppose two maps h, k : A → B represent some morphisms. They rep-resent the same morphism if and only if

∀a ∈ A, α(a) ≤ β(ha, ka).

(4) Let (C, γ) be another O(X)-valued set and ψ : (B, β)→ (C, γ) be anothermorphism. If ϕ (resp. ψ) is represented by a map h (resp. k), then ψϕ isrepresented by kh.

Proposition 1.11. Let ϕ : (A,α)→ (B, β) be a morphism represented by h.

• ϕ is monic ⇐⇒ ∀a, a′ ∈ A, α(a, a′) = α(a) ∧ α(a′) ∧ β(ha, ha′).

• ϕ is epic ⇐⇒ ∀b ∈ B, β(b) =∨a[α(a) ∧ β(ha, b)].

Further, if α(a) = β(ha) for all a ∈ A, these conditions reduce to

• ϕ is monic ⇐⇒ ∀a, a′ ∈ A, α(a, a′) = β(ha, ha′).

• ϕ is epic ⇐⇒ ∀b ∈ B, β(b) =∨a β(ha, b).

Combining the above facts, we obtain

Corollary 1.12. If h satisfies α(a) = β(ha) for all a ∈ A, then h represents anisomorphism exactly when the following conditions hold

∀a, a′ ∈ A, α(a, a′) = β(ha, ha′), and ∀b ∈ B, β(b) =∨a

β(ha, b).

If h satisfies these conditions, the induced isomorphism ϕ(a, b) = β(ha, b) hasthe inverse ϕ−1(b, a) = β(ha, b).

1.2 Sheaves on Locales and Complete Heyting-Valued Sets

Continuing from the previous section, we fix a locale X . Let us discuss therelationship between sheaves on X and O(X)-valued sets. Our presentationstyle here is largely due to [Elephant, §C1.3].

Definition 1.13.

(1) A presheaf on X is a functor O(X)op → Set. For a presheaf P andU ∈ O(X), elements of PU (resp. of P1X ) are called sections of P on U

(resp. global sections of P ). If a ∈ PU and W ≤ U , we will write a|Wfor P (W ≤ U)(a).

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(2) A presheaf P is said to be a sheaf on X when it satisfies the followingcondition: For any covering Uii of U ∈ O(X) (i.e., U =

∨i Ui) and any

family aii of sections ai ∈ PUi, if ai|Ui∧Uj = aj |Ui∧Uj for all i, j, thenthere exists a unique a ∈ PU such that a|Ui = ai for all i.

(3) Morphisms of presheaves are defined to be natural transformations.The functor category SetO(X)op is also called the category of presheaves.Let Sh(X) denote its full subcategory spanned by sheaves.

We can associate a presheaf P with anO(X)-valued set Θ(P ) := (∐U PU, δP )

as follows: for (a, b) ∈ PU × PV ⊆∐U PU ×

∐U PU ,

δP (a, b) :=∨W ≤ U ∧ V ; a|W = b|W .

Notice that

• a ∈ PU if and only if δP (a) = U . Hence, global elements of Θ(P ) areexactly global sections of P .

• If P is a sheaf, then δP (a, b) is the largest element on which the restric-tions of a and b coincide. Moreover, if X is a topological space,

δP (a, b) = x ∈ X ; ax = bx ,

where ax, bx are the germs of a, b over x.

For a morphism ξ : P → Q of presheaves onX , the induced map h :∐U PU →∐

U QU satisfies

∀(a, b) ∈(∐

U

PU

)2

, δP (a, b) ≤ δQ(ha, hb) and δP (a) = δQ(ha).

Therefore, by Proposition 1.10, h represents a morphism Θ(ξ) : Θ(P )→ Θ(Q).This construction gives a functor Θ: SetO(X)op → Set(O(X)).

Notice that a presheaf P is separated if and only if, for any a, b, δP (a) =

δP (b) = δP (a, b) implies a = b. Fourman & Scott [FS79] say Heyting-valuedsets are separated if the latter condition holds. To give a similar characteriza-tion of sheaves, we need a more involved definition.

Definition 1.14. For an O(X)-valued set (A,α), define a preorder v on A by

b v a def.⇐==⇒ α(a, b) = α(b).

(A,α) is said to be complete if the following conditions hold:

• v is a partial order. (This is equivalent to separatedness.)

• For any a ∈ A and U ≤ α(a), there exists b ∈ A such that b v a andα(b) = U . (If v is a partial order, such b is uniquely determined anddenoted by a|U .)

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• If a family aii of elements of A is pairwise compatible, i.e., α(ai, aj) =

α(ai)∧α(aj) for all i, j, then it has a supremum w.r.t. v. (The supremumis called an amalgamation of aii.)

Let CSet(O(X)) denote the full subcategory of Set(O(X)) spanned by com-plete O(X)-valued sets.

Proposition 1.15. A presheaf P on X is a sheaf if and only if Θ(P ) is completeas an O(X)-valued set. Moreover, for any complete O(X)-valued set (A,α),there exists a sheaf P on X such that (A,α) and Θ(P ) are isomorphic.

Proof. The latter part: if (A,α) is complete, then by putting

PU := a ∈ A ; α(a) = U ,

we have a desired sheaf P on X .

We can rephrase completeness in terms of singletons.

Definition 1.16. Let (A,α) be an O(X)-valued set. A singleton on (A,α) is afunction σ : A→ O(X) such that

∀a, a′ ∈ A, σ(a) ∧ α(a, a′) ≤ σ(a′) and σ(a) ∧ σ(a′) ≤ α(a, a′).

In particular, σ(a) ≤ α(a) always holds.

For each a ∈ A, the map σa := α(a,−) is a singleton of (A,α).

Lemma 1.17. For an O(X)-valued set (A,α), TFAE:

(i) (A,α) is complete.

(ii) Any singleton of (A,α) is of the form σa for a uniquely determined a.

Proof. (i)⇒(ii): Suppose (A,α) is complete. Let σ be a singleton on (A,α).Then the family

a|σ(a) ; a ∈ A

is pairwise compatible, and its supremum

s ∈ A satisfies σ = σs.(ii)⇒(i): Suppose the condition (ii) holds. If α(a) = α(a′) = α(a, a′), thenσa = σa′ and hence a = a′. Thus v is anti-symmetric. If a ∈ A and U ≤ α(a),the map α(a,−) ∧ U is a singleton, and we then have the restriction a|U . Ifa family aii is pairwise compatible, the map

∨i α(ai,−) is a singleton, and

we then have the amalgamation.

Lemma 1.18. Let (A,α), (B, β) be O(X)-valued sets with (B, β) complete.Each morphism ϕ : (A,α)→ (B, β) is represented by a unique map h : A→ B

which satisfies α(a, a′) ≤ β(ha, ha′) and α(a) = β(ha) for all a, a′ ∈ A.

Proof. For any fixed a ∈ A, the map ϕ(a,−) is a singleton of (B, β). By com-pleteness, we can find a unique ha ∈ B such that ϕ(a, b) = β(ha, b) for everyb ∈ B. This defines a map h : A → B representing ϕ and having the desiredproperties.

This lemma and Proposition 1.10(4) yield

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Proposition 1.19. Θ induces a categorical equivalence between Sh(X) andCSet(O(X)).

On the other hand, we can also show that Set(O(X)) and CSet(O(X)) arecategorically equivalent.

Proposition 1.20. Let A be the set of singletons on (A,α). Define a valuationα on A by

α(σ, τ) :=∨a∈A

σ(a) ∧ τ(a).

Then (A, α) is a complete O(X)-valued set, and the map A × A 3 (σ, a) 7→σ(a) ∈ O(X) represents an isomorphism (A,α) ' (A, α). We call (A, α) thecompletion of (A,α).

For a morphism ϕ : (A,α)→ (B, β), let ϕ be the composite of

(A, α)∼→ (A,α)

ϕ−→ (B, β)∼→ (B, β).

Then ˜(−) : Set(O(X)) → CSet(O(X)) becomes a functor and also gives aquasi-inverse of the inclusion functor.

Corollary 1.21. The categories Sh(X), Set(O(X)) and CSet(O(X)) are cate-gorically equivalent. In particular, Set(O(X)) is a Grothendieck topos.

1.3 Topos Structure of Set(O(X))

In the previous section, we saw that Set(O(X)) is a Grothendieck topos. Here,we will give a concrete description of the topos structure of Set(O(X)). Mostresults here (except for some details on the lattice P(A,α)) are borrowed from[Hig84]. The constructions will be exploited later in this paper.

Proposition 1.22 (Finite limits in Set(O(X))).

(1) Let (∗,>) be the O(X)-valued set with >(∗, ∗) = 1X . This yields aterminal object in Set(O(X)).

(2) Let (Ai, αi)i∈I be a finite family of O(X)-valued sets. Define a val-uation δ on

∏iAi by δ(a, a′) =

∧i αi(ai, a

′i) for a = aii∈I and a′ =

a′ii∈I . Then (∏iAi, δ) equipped with the canonical projections (

∏iAi, δ)→

(Ai, αi) is a product of (Ai, αi)i∈I in Set(O(X)).

(3) Let ϕ,ψ : (A,α) ⇒ (B, β) be morphisms of O(X)-valued sets. Define avaluation δ on A by δ(a, a′) = α(a, a′)∧

∨b∈B ϕ(a, b)∧ψ(a, b). Then (A, δ)

equipped with the canonical morphism (A, δ) (A,α) is an equalizerof ϕ and ψ in Set(O(X)).

(4) Let (A,α)ϕ−→ (C, γ)

ψ←− (B, β) be morphisms ofO(X)-valued sets. Definea valuation δ on A×B by

δ((a, b), (a′, b′)) = α(a, a′) ∧ β(b, b′) ∧∨c∈C

ϕ(a, c) ∧ ψ(b, c).

Then (A×B, δ) equipped with the canonical projections is a pullback ofthat diagram in Set(O(X)).

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The following notion of strict relation is crucial in handling subobjects ofan O(X)-valued set. In the next section, it will enable us to define the “forcingvalues” of formulas.

Definition 1.23. Let (A,α) be an O(X)-valued set. A strict relation on (A,α)

is a function σ : A→ O(X) such that

∀a, a′ ∈ A, σ(a) ∧ α(a, a′) ≤ σ(a′) and σ(a) ≤ α(a).

Note that a singleton is a strict relation on the same O(X)-valued set.

Proposition 1.24. Let P(A,α) be the set of strict relations on (A,α) orderedby

σ ≤ τ def.⇐==⇒ ∀a ∈ A, σ(a) ≤ τ(a).

Then, as ordered sets, P(A,α) is isomorphic to the poset Sub(A,α) of subob-jects of (A,α).

Proof. For a strict relation σ, we define a map ασ : A×A→ O(X) by

ασ(a, a′) := σ(a) ∧ α(a, a′) = σ(a′) ∧ α(a, a′).

Then (A,ασ) is an O(X)-valued set and the identity map on A represents amonomorphism ισ : (A,ασ) (A,α) by Proposition 1.11.

Conversely, for a monomorphism ϕ : (B, β) (A,α), we define a strictrelation ρϕ : A→ O(X) by

ρϕ(a) :=∨b∈B

ϕ(b, a).

Then we can check

• for any σ, ρ(ισ) = σ.

• for any ϕ, ι(ρϕ) ' ϕ as subobjects of (A,α).

In fact, for an arbitrary morphism ϕ, a strict relation ρϕ can be defined asabove, and the image factorization of ϕ is given by

(B, β) (A,α)

(A,α(ρϕ))

ϕ

ϕ ι(ρϕ)

.

Set(O(X)) is a topos and, in particular, a Heyting category ( [Elephant,§A1.4]). The associated operations on subobject lattices are as follows:

Proposition 1.25. The operations on the frame P(A,α) are given by

1P(A,α)(a) = α(a), 0P(A,α)(a) = 0X ,

(σ ∧ τ)(a) = σ(a) ∧ τ(a),(∨i

σi

)(a) =

∨i

σi(a),

(σ → τ)(a) = α(a) ∧ (σ(a)→ τ(a)).

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Proposition 1.26. Let ϕ : (B, β) → (A,α) be a morphism. Pulling back sub-objects along ϕ defines a frame homomorphism ϕ∗ : P(A,α) → P(B, β) suchthat, for σ ∈ P(A,α),

(ϕ∗σ)(b) =∨a∈A

ϕ(b, a) ∧ σ(a) = β(b) ∧∧a∈A

[ϕ(b, a)→ σ(a)].

Proof. For the last identity, note that

β(b) ∧∧a∈A

[ϕ(b, a)→ σ(a)] =∨a∈A

ϕ(b, a) ∧∧a′∈A

[ϕ(b, a′)→ σ(a′)]

≤∨a∈A

ϕ(b, a) ∧ [ϕ(b, a)→ σ(a)]

≤∨a∈A

ϕ(b, a) ∧ σ(a).

Proposition 1.27. In the same notations as above, ϕ∗ has both a left adjoint ∃ϕand a right adjoint ∀ϕ: for τ ∈ P(B, β),

(∃ϕτ)(a) =∨b∈B

ϕ(b, a) ∧ τ(b),

(∀ϕτ)(a) = α(a) ∧∧b∈B

[ϕ(b, a)→ τ(b)].

Finally, we describe the higher-order structure of Set(O(X)).

Proposition 1.28. Put δ(U, V ) = (U → V ) ∧ (V → U) for U, V ∈ O(X). Then(O(X), δ) is an O(X)-valued set. Let t : (∗,>)→ (O(X), δ) be the morphismdefined by t(∗, U) = U . This yields a subobject classifier of Set(O(X)).

Proof. Let χ : (A,α) → (O(X), δ) be a morphism. Since t corresponds tothe strict relation idO(X) on (O(X), δ), the pullback of t along χ is givenby the strict relation σ(a) =

∨U χ(a, U) ∧ U . Conversely, given a strict re-

lation σ on (A,α), then σ itself represents a morphism χ(a, U) = α(a) ∧(U ↔ σ(a)). These correspondences yield a bijection between P(A,α) andHom((A,α), (O(X), δ)).

(A,ασ)

(A,α)

(∗,>)

(O(X), δ)

y

!

tισ

χ

Similarly to O(X), P(A,α) is not only a frame but also an O(X)-valuedset.

Proposition 1.29. The power object of (A,α) is given by P(A,α) equippedwith the valuation 2

α(σ, τ) :=∧a∈A

σ(a)↔ τ(a).

2Note that α does not necessarily coincide with α on A in Proposition 1.20. Indeed, whileα(σ, τ) ≤ α(σ, τ) holds for any σ, τ ∈ A, the converse inequality does not hold if σ = τ = σa fora non-global element a ∈ A.

11

Proof. We would like to establish the following bijection

Hom((B, β),P(A,α)) ' P((B, β)× (A,α)).

For a morphism ϕ : (B, β) → P(A,α), we have a strict relation θ on (B, β) ×(A,α) such that

θ(b, a) =∨

τ∈P(A,α)

ϕ(b, τ) ∧ τ(a).

On the other hand, for any strict relation θ, we have a morphism

ϕ(b, τ) = β(b) ∧∧a∈A

θ(b, a)↔ τ(a).

These correspondences are mutual inverses.

2 Sheaves of Structures and Heyting-Valued Struc-tures

2.1 Structures in a Topos

We will be concerned with categorical semantics in the toposes Sh(X) andSet(O(X)). In this subsection, we take a glance at first-order categorical logic,which originates from [MR77]. The main reference is [Elephant], in particular,Chapter D1 in volume 2. For an overview, we refer the reader to Caramello’saccount [Cara14], which is a preliminary version of the first two chaptersof her book [TST]. Although many fragments of (possibly infinitary) first-order logic are considered in the context of categorical logic, we restrict ourattention to single-sorted intuitionistic logic. Examples of other fragmentsinclude Horn, cartesian, regular, coherent, classical, and geometric logics.

Definition 2.1. A (first-order) language L consists of the following data:

• A set L-Func of function symbols. Each function symbol f ∈ L-Func isassociated with a natural number n (the arity of f ). If n = 0, f is calleda constant (symbol).

• A set L-Rel of relation symbols. Each relation symbol R ∈ L-Rel isassociated with a natural number n (the arity of R). If n = 0, R is calledan atomic proposition.

L-terms and L-formulas are defined as usual. We need some conventions.

Definition 2.2.

(1) A context is a finite list u ≡ u1, . . . , un of distinct variables. If n = 0, it iscalled the empty context and denoted by [ ].

(2) We say that a context u is suitable for an L-formula ϕ when u containsall the free variables of ϕ. A formula ϕ equipped with a suitable contextu is called a formula-in-context and indicated by ϕ(u). Similarly, terms-in-context can be defined.

12

For an L-formula-in-context ϕ(u,v), we abbreviate, e.g., ∃v1 · · · ∃vnϕ(u,v)

as ∃vϕ(u,v). We also abbreviate the L-formula∧i ui = vi as u = v, where

u,v are assumed to have the same length. A formula is closed if it containsno free variables.

We now give categorical semantics in an arbitrary elementary topos E ,though we will only need the case when E is a Grothendieck topos.

Definition 2.3. Let L be a language and E a topos. An L-structureM in E isgiven by specifying the following data:

• We have the underlying object |M| ∈ E and denote the n-ary product by|M|n. In particular, |M|0 is the terminal object 1E .

• To an n-ary function symbol f , we assign a morphism fM : |M|n →|M|.

• To an n-ary relation symbol R, we assign a subobject RM of |M|n.

As usual, we will not distinguishM and its underlying object |M| in notation.

Interpretations of L-terms and L-formulas are defined by using internaloperations in E .

Definition 2.4 (Interpretations of terms). Let M be an L-structure in a toposE . For an L-term-in-context t(u), we define the interpretation tM : Mn →Minductively.

• If t is a variable ui, then tM is the i-th product projection πi : Mn →M.

• If interpretations of L-terms ti(u) and s(v) are given, the term s(t1(u), . . . , tm(u))

is interpreted as the composite of the following morphisms:

Mn 〈tM1 ,...,tMm 〉−−−−−−−→Mm sM−−→M,

where 〈tM1 , . . . , tMm 〉 is the morphism obtained from the morphisms tMiby using the universal property of the productMm.

Definition 2.5 (Interpretations of formulas). Let M be an L-structure in atopos E . For an L-formula-in-context ϕ(u), we define the interpretation Ju. ϕKMas a subobject of Mn inductively. (We drop the subscript M if no confusionarises.)

• If ϕ ≡ (s(u) = t(u)) where s, t are terms, then Ju. ϕK is defined to be theequalizer of

Mn MsM

tM .

• If ϕ ≡ R(t1(u), . . . , tm(u)), then Ju. ϕK is the pullback

Ju. R(t1, . . . , tm)K

Mn

RM

Mm〈tM1 , . . . , tMm 〉

.

13

• If ϕ ≡ >,⊥, ψ ∧ θ, ψ ∨ θ, ψ → θ or ¬ψ, then Ju. ϕK is defined as expectedby using the Heyting operations on Sub(Mn).

• If ϕ ≡ ∃vψ(u, v), then Ju. ϕK is the image as in the following diagram:

Ju, v. ψK

Ju.∃vψK

Mn ×M

Mn

π

,

where π is the projection ontoMn.

• If ϕ ≡ ∀vψ(u, v), then Ju. ϕK := ∀π Ju, v. ψK, where ∀π : Sub(Mn×M)→Sub(Mn) is the right adjoint of π∗.

In this paper, we will not consider the notions of models of a theory in atopos nor homomorphisms between structures.

2.2 Sheaves of Structures and Heyting-Valued Structures

We now investigate the relationship between structures in Sh(X) and thosein Set(O(X)). We first consider the case of sheaves on a topological spaceX . Let LH be the category of topological spaces and local homeomorphismsbetween them. Recall that the slice category LH/X is categorically equivalentto Sh(X). Comer [Com74], Ellerman [Ell74], and Macintyre [Maci73] used thefollowing notion to obtain model-theoretic results:

Definition 2.6. A sheaf of L-structures (on X) is a tuple(X,E, π,

fEx ; x ∈ X, f ∈ L-Func

,REx ; x ∈ X, R ∈ L-Rel

)such that

• π : E → X is a local homeomorphism of topological spaces,

• each stalk Ex equipped with fExf and RExR is an L-structure, and

– for each function symbol f , the map∐x(Ex)n → E induced by

fExx is continuous,

– for each relation symbol R, the subset∐xR

Ex ⊆∐x(Ex)n is open,

where∐x(Ex)n is seen as a subspace of the product space En for n > 0,

and∐x(Ex)0 ' X .

Sheaves of abelian groups or of rings in geometry are, of course, suchexamples for suitable languages. We will meet other examples which givemodel-theoretic constructions of (usual Set-valued) structures in the next sec-tion.

Lemma 2.7. A sheaf of L-structures is identified with an L-structure in LH/X .

Proof. Notice the following facts:

14

•∐x(Ex)n is a fiber product E ×X · · · ×X E, i.e., a product in LH/X .

• Any monomorphism in LH/X is an open embedding.

Hereafter, we fix a locale X . We will also say “sheaves of structures on X”to mean structures in Sh(X). When we mention a subsheaf Q of a sheaf P ,each Q(U) is assumed to be a subset of P (U).

Before we define Heyting-valued structures, let us introduce space-savingnotations. If (M, δ) is an O(X)-valued set, then the n-th powerMn is canoni-cally equipped with the valuation as in Proposition 1.22(2). For tuples a,a′ ∈Mn, we simply write δ(a,a′) (resp. δ(a)) for

∧i δ(ai, a

′i) (resp.

∧i δ(ai)). These

notations are useful, but, in the case n = 2, we will always write δ(a)∧ δ(b) forδ((a, b), (a, b)) to avoid confusion between δ(a, b) and δ((a, b)).

Definition 2.8. An O(X)-valued L-structure is an L-structure in the toposSet(O(X)), i.e., it consists of the following data:

• an O(X)-valued set (M, δ),

• for each function symbol f , a morphism fM : (Mn, δ)→ (M, δ),

• for each relation symbol R, a strict relation RM : Mn → O(X).

The interpretation of equality is the diagonal (M, δ) (M2, δ), which corre-sponds to the strict relation (a, b) 7→ δ(a, b) on (M2, δ) under the bijection inProposition 1.24.

Fourman & Scott [FS79, p. 365] defined Heyting-valued structures in aslightly less general form. Structures in the topos associated with a tripos arediscussed in [vOos08, p. 69].

Recall the construction of Θ: SetO(X)op → Set(O(X)) at the beginning of§1.2. We can obtain O(X)-valued structures from sheaves of structures on X

by applying Θ.

Lemma 2.9. Let P be a presheaf on X . Then, the n-ary product Θ(P )n is iso-morphic to Θ(Pn) asO(X)-valued sets. Indeed, the canonical map h :

∐U (PU)n →

(∐U PU)n represents an isomorphism ι : Θ(Pn)

∼→ Θ(P )n so that ι(b,a) =

ι−1(a, b) = δP (h(b),a).Moreover, for a strict relation σ on Θ(Pn), the corresponding strict relation

τ on Θ(P )n is given by

τ(a) =∨

b∈Θ(Pn)

σ(b) ∧ δP (h(b),a).

Proof. For the case when P is a sheaf, this lemma is an immediate conse-quence of the fact that Θ: Sh(X) → Set(O(X)) is part of an equivalence ofcategories. We can also see directly that Θ: SetO(X)op → Set(O(X)) preservesfinite products by using Corollary 1.12.

For a given σ, by the proof of Proposition 1.24, the corresponding sub-object of Θ(Pn) is (

∐U (PU)n, (δPn)σ) with (δPn)σ(b, b′) = σ(b) ∧ δPn(b, b′).

Hence, τ is given by

τ(a) =∨

b,b′∈Θ(Pn)

(δPn)σ(b, b′) ∧ ι(b′,a) =∨

b∈Θ(Pn)

σ(b) ∧ δP (h(b),a).

15

We remark that τ(h(b)) = σ(b) for any b ∈ Θ(Pn) and therefore τ is anextension of σ along h.

Proposition 2.10. If P is a sheaf of L-structures on X , then we can make theO(X)-valued set Θ(P ) into an O(X)-valued L-structure canonically.

Proof. Here we describe in detail the corresponding O(X)-valued L-structureM. For each function f , we have a morphism fP : Pn → P of sheaves. Thisinduces a morphism Θ(fP ) : Θ(Pn) → Θ(P ). By the previous lemma, weobtain a morphism fM : Θ(P )n → Θ(P ), which can be computed as

fM(a, a′) =∨

b∈Θ(Pn)

ι−1(a, b) ∧Θ(fP )(b, a′) = δP (fP (a|δP (a)), a′)

for a ∈ Θ(P )n and a′ ∈ Θ(P ), where a|δP (a) = (a1|δP (a), . . . , an|δP (a)). In par-ticular, fM is represented by the map k : Mn →M with k(a) = fP (a|δP (a)).

For each relation R, we have a subsheaf RP Pn. This induces a sub-object Θ(RP ) Θ(Pn), which corresponds to the following strict relationσ :∐U (PU)n → O(X): for b ∈ (PU)n,

σ(b) =∨

b′∈Θ(RP )

δPn(b, b′) =∨

W ≤ U ; b|W ∈ RP (W ).

By the previous lemma, we obtain a subobject of Θ(P )n, which correspondsto the following strict relation RM : Mn → O(X): for a ∈ PU1 × · · · × PUn,

RM(a) =∨

b∈Θ(Pn)

ι−1(a, b) ∧ σ(b)

=∨

b∈Θ(Pn)

[δP (h(b),a) ∧

∨W ≤ δPn(b) ; b|W ∈ RP (W )

]=∨

W ≤ U1 ∧ · · · ∧ Un ; a|W ∈ RP (W ).

Notice that the subobject Θ(P ) Θ(P )2 obtained from the diagonal P P 2 is the same as the one determined by the strict relation δP on Θ(P ).

We could describe the converse construction (from Heyting-valued struc-tures to sheaves of structures). This involves a complicated use of completionof Heyting-valued sets, and we do not find such details to be useful for thepurpose of this paper. So we skip it at this point.

In the context of set theory (e.g. [PV20]), there are examples of Heyting-valued structures which do not come from sheaves.

Remark 2.11. Some authors have applied (set-theoretic) Boolean-valued uni-verses to mathematics (cf. [KK99] and the references therein). From the view-point of Remark 1.6, these works complement our understanding of Heyting-valued structures.

16

2.3 Forcing Values of Formulas

Forcing values of formulas derive from Boolean-valued set theory. Here wefirst define them categorically and then observe that our definition is com-patible with the usual one. The categorical description seems to be folklorebut has not appeared in an explicit form elsewhere. For an O(X)-valued L-structure (M, δ), we write LM for the language extending L by adding a newconstant symbol for each element ofM.

Definition 2.12. For an L-formula-in-context ϕ(u), the strict relation ‖ϕ(−)‖M

on (M, δ)n is defined to be the one corresponding to the subobject Ju. ϕK(M,δ)

(M, δ)n. For a ∈ Mn, ‖ϕ(a)‖M is called the forcing value of the closed LM-formula ϕ(a). We drop the superscriptM if no confusion arises.

Since the strict relation a 7→ δ(a) is the greatest element in P(Mn, δ),‖ϕ(a)‖M ≤ δ(a) always holds. Using the results in §1.3, we can calculate theforcing values inductively.

Proposition 2.13.

‖R(t1(a), . . . , tm(a))‖M =∨

b∈Mm

〈tM1 , . . . , tMm 〉(a, b) ∧RM(b),

‖s(a) = t(a)‖M =∨

b,c∈M

〈sM, tM〉(a, (b, c)) ∧ δ(b, c),

‖ϕ(a) ∧ ψ(a)‖M = ‖ϕ(a)‖M ∧ ‖ψ(a)‖M ,

‖ϕ(a) ∨ ψ(a)‖M = ‖ϕ(a)‖M ∨ ‖ψ(a)‖M ,

‖ϕ(a)→ ψ(a)‖M = δ(a) ∧[‖ϕ(a)‖M → ‖ψ(a)‖M

],

‖∃vϕ(a, v)‖M =∨b∈M

‖ϕ(a, b)‖M ,

‖∀vϕ(a, v)‖M = δ(a) ∧∧b∈M

[δ(b)→ ‖ϕ(a, b)‖M

].

Remark 2.14. If a formula ϕ has a suitable context u and v is a variable dis-tinct from u, we have to distinguish the formulas-in-context ϕ(u) and ϕ(u, v).Indeed, the forcing values ‖ϕ(a)‖ and ‖ϕ(a, b)‖ can be different and

‖ϕ(a, b)‖ = ‖ϕ(a)‖ ∧ δ(b).

This description of forcing values is compatible with those in [FS79, Defini-tion 5.13], [TvD88, Definition 13.6.6] and [vOos08, p. 70]. The soundness andcompleteness theorems for Heyting-valued semantics are usually formulatedwith respect to intuitionistic predicate logic with existence predicate (for short,IQCE) as in [TvD88, §2.2, §13.6]. However, we will only need soundness ofthe following form:

Lemma 2.15. If the sentence ∀u[ϕ(u) → ψ(u)] is intuitionistically valid, then‖ϕ(a)‖M ≤ ‖ψ(a)‖M ≤ δ(a) holds for any a ∈Mn.

17

Proof. The assumption implies Ju. ϕK ≤ Ju. ψK as subobjects of (M, δ)n. There-fore, the conclusion holds by the definition of forcing values.

Let P be a sheaf of L-structures and Θ(P ) = (M, δ) be the O(X)-valuedL-structure obtained from Proposition 2.10. We can see

(1) For any L-term t(u), the morphism tM is represented by the mapMn 3a 7→ tP (a|δ(a)) ∈M where tP : Pn → P is the interpretation of t by P .

(2) For any atomic L-formula R(t1(u), . . . , tm(u)) and a ∈Mn,

‖R(t1(a), . . . , tm(a))‖M

= RM(tP1 (a|δ(a)), . . . , tPm(a|δ(a)))

=∨

W ≤ δ(a) ; (tP1 (a|W ), . . . , tPm(a|W )) ∈ RP (W ).

Similarly for the formula s(u) = t(u).

More generally, the forcing value ‖ϕ(−)‖ for Θ(P ) can be described interms of the subsheaf Ju. ϕK of Pn. Let Ω be the sheaf U 7→ Ω(U) = (U)↓.This is a subobject classifier in Sh(X), and we thus obtain the characteristicmorphism χ : Pn → Ω by the universality of the subobject classifier:

Ju. ϕKy

1

Pn Ω

!

χ

true χU (a) =∨W ≤ U ; a|W ∈ Ju. ϕK (W ) .

Using Proposition 1.28 and the fact that Θ(Ω) and (O(X), δ) in that propo-sition are canonically isomorphic, we can verify the following:

Proposition 2.16 (definable subsheaves and forcing values). In the above no-tation, χU (a) = ‖ϕ(a)‖M for any a ∈ PnU . We will denote χ by ‖ϕ(−)‖P andits component χU by ‖ϕ(−)‖PU .

Let y : O(X)→ SetO(X)op be the Yoneda embedding, and a : SetO(X)op →Sh(X) the associated sheaf functor. We write a : ayU → Pn for the morphismcorresponding to a ∈ PnU under the bijection

PnU ' HomSetO(X)op (yU,Pn) ' HomSh(X)(ayU,Pn).

In terms of forcing values, the sheaf semantics in Sh(X) (cf. [SGL, §VI.7]) hasa simple description:

U P ϕ(a)

def.⇐==⇒ the morphism a : ayU → Pn factors through the subsheaf Ju. ϕK Pn,

⇐⇒ ‖ϕ(−)‖P a = true !,

⇐⇒ ‖ϕ(a)‖PU = U.

This is the reason why we use the term “forcing values” similarly as in [Ell74].Using the above description, we can show the properties of forcing relation[SGL, Theorem VI.7.1] for the usual site on O(X).

18

3 Filter-Quotients of Heyting-Valued Structures andŁos’s Theorem

As we promised after Definition 2.6, we will observe that sheaves of struc-tures give some constructions in model theory. These constructions can begeneralized to constructions for Heyting-valued structures, and they providean adequate setup to state our Łos-type theorem.

3.1 Model-Theoretic Constructions via Sheaves of Structures

Definition 3.1. Let P be a sheaf of L-structures on a locale X .

(1) We make the set P (U) for a fixed U into an L-structure as follows:

fP (U)(a) := (fP )U (a), and P (U) |= R(a)def.⇐==⇒ a ∈ RP (U) ⊆ P (U)m.

(2) For a filter f on O(X), the colimit P/f := lim−→U∈fP (U) is the quotient

of∐U∈f P (U) by the following equivalence relation: for U, V ∈ f and

a ∈ P (U), b ∈ P (V ),

(U, a) ∼ (V, b)def.⇐==⇒ ∃W ∈ f, W ≤ U ∧ V and a|W = b|W .

We often write [a]f for a tuple ([a1]f, . . . , [an]f) of equivalence classes. Letδ be the valuation of Θ(P ). We make P/f into an L-structure as follows:

fP/f([a]f) := [fP (a1|δ(a), . . . , an|δ(a))]f,

P/f |= R([a]f)def.⇐==⇒ ∃W ∈ f, a|W ∈ RP (W ),

⇐⇒ ∃W ∈ f, P (W ) |= R(a|W ).

In particular, if X is a topological space and x ∈ X , each stalk Px is thequotient P/nx by the filter nx of open neighborhoods of x.

Example 3.2 (Products). Let X be a set. Given an X-indexed family Mxx∈Xof L-structures, the product N :=

∏x∈XMx is an L-structure such that, for

any elements ai = aixx∈X ,

fN (a1, . . . , an) :=fMx(a1

x, . . . , anx)x∈X ,

N |= R(a1, . . . , an)def.⇐==⇒ ∀x ∈ X,Mx |= R(a1

x, . . . , anx).

Giving an X-indexed family of L-structures is the same as giving a sheaf ofL-structures on the discrete space X . Let P be the sheaf corresponding tothe local homeomorphism

∐x∈XMx → X given by the canonical projection.

Then, the L-structure P (X) of global sections is the same as N .Notice that, by induction based on Proposition 2.13,∥∥ϕ(a1, . . . , an)

∥∥Θ(P )=x ∈ X ;Mx |= ϕ(a1

x, . . . , anx)

holds for any formula ϕ and a1, . . . , an ∈ N .

19

Example 3.3 (Ultraproducts). Let u be an ultrafilter over a set X . In the samenotation as the previous example, the ultraproduct

∏xMx/u is the quotient

of∏xMx by the equivalence relation

a ∼ b def.⇐==⇒ x ∈ X ; ax = bx ∈ u

equipped with canonical interpretations of L, e.g.,∏x

Mx/u |= R([a1]u, . . . , [an]u)

def.⇐==⇒x ∈ X ;Mx |= R(a1

x, . . . , anx)∈ u.

If eachMx is non-empty,∏xMx/u can be described as a filter-quotient of the

sheaf P corresponding to∐x∈XMx → X . Since P (U) =

∏x∈UMx and each

local section can be extended to a global section by non-emptiness, we have∏x

Mx/u ' lim−→U∈u

P (U) = P/u.

Thus, it is reasonable to regard P/u as a “generalized” ultraproduct for any u

(cf. §3.3.1). Notice that we need the axiom of choice to extend local sections toglobal ones, but we do not need AC if L contains a constant symbol.

Example 3.4 (Bounded Boolean Powers). Let B be a Boolean algebra and Mbe an L-structure. We then have the sheaf P on the Stone space X dual to Bdetermined by

P (U) := s : U →M ; locally constant map .

This becomes a sheaf of L-structures, and M[B]ω := P (X) is said to be thebounded Boolean power ofM (cf. [Hod93, §9.7]).

Example 3.5 (Bounded Boolean Ultrapowers). In the same notation as theprevious example, for any s, t ∈M[B]ω , the subsets

‖R(s1, . . . , sn)‖ = v ∈ X ;M |= R(s1(v), . . . , sn(v)) ,‖s = t‖ = v ∈ X ; s(v) = t(v)

are clopen and identified with elements of B. Let u be an ultrafilter on B (= apoint of X). The bounded Boolean ultrapowerM[B]ω/u is given by

s ∼ t def.⇐==⇒ ‖s = t‖ ∈ u,

M[B]ω/u |= R([s1]u, . . . , [sn]u)def.⇐==⇒ ‖R(s1, . . . , sn)‖ ∈ u.

M[B]ω/u has a representation as a filter-quotient

M[B]ω/u ' lim−→U∈u

P (DU ) ' Pu,

where DU = v ∈ X ; U ∈ v and Pu is the stalk over u.

Bounded Boolean (ultra)powers are not direct generalizations of ordinary(ultra)powers. Unbounded Boolean (ultra)powers are such things, while theyinvolve more complicated sheaf-theoretic constructions. Fish [Fis00] gives asurvey of bounded and unbounded Boolean (ultra)powers. These construc-tions can be further generalized to the notion of Boolean product (see [BW79],[Wer82], and [BS12]), which involves sheaves on Stone spaces.

20

3.2 Filter-Quotients of Heyting-Valued Structures

We will generalize the construction of P/f to Heyting-valued structures. Weuse filter-quotients of Heyting-valued sets (or structures), which appeared in,e.g., [PV20, Definition 2.6] and [Mir20, Chapter 34]. Let (M, δ) be an O(X)-valued L-structure. Given a filter f on O(X), an (O(X)/f)-valued L-structureM/f is defined as follows: 3 we first observe

Claim. The following relation ∼f onM is an equivalence relation

a ∼f bdef.⇐==⇒ [δ(a) ∨ δ(b)→ δ(a, b)] ∈ f.

Proof. For transitivity, observe

(δ(a) ∨ δ(b)→ δ(a, b)) ∧ (δ(b) ∨ δ(c)→ δ(b, c)) ∧ (δ(a) ∨ δ(c))= [δ(a) ∧ (δ(a) ∨ δ(b)→ δ(a, b)) ∧ (δ(b) ∨ δ(c)→ δ(b, c))]

∨ [δ(c) ∧ (δ(a) ∨ δ(b)→ δ(a, b)) ∧ (δ(b) ∨ δ(c)→ δ(b, c))]

(by using δ(a, b) ≤ δ(a) ∧ δ(b) etc.,)

= δ(a, b) ∧ δ(b, c) ≤ δ(a, c).

We then have

(δ(a) ∨ δ(b)→ δ(a, b)) ∧ (δ(b) ∨ δ(c)→ δ(b, c)) ≤ δ(a) ∨ δ(c)→ δ(a, c).

We denote the quotientM/∼f byM/f and the equivalence class of a ∈Mby [a]f. In particular, by applying this to the O(X)-valued set (O(X),∧), forwhich U ∼f V iff (U ↔ V ) ∈ f, we have the quotient Heyting algebra O(X)/f.By defining the valuation 4

δf([a]f, [b]f) := [δ(a, b)]f,

we can makeM/f into an (O(X)/f)-valued set except thatO(X)/f is not neces-sarily complete. We may use the Dedekind–MacNeille completion of O(X)/f

(cf. [Joh82, III.3.11]) to define forcing values as in [PV20, Definition 2.2], butsuch a complication will not be necessary for this paper because we will useforcing values ‖ϕ([a]f)‖M/f only for atomic formulas ϕ.

For each function f and each relationR, the morphism fM/f : ((M/f)n, δf)→(M/f, δf) and the strict relation RM/f : (M/f)n → O(X)/f are defined canoni-cally:

fM/f([a1]f, . . . , [an]f, [b]f) := [fM(a1, . . . , an, b)]f,

RM/f([a1]f, . . . , [an]f) := [RM(a1, . . . , an)]f.

We have finished the construction of the (O(X)/f)-valued L-structure M/f.We will call it the filter-quotient ofM by f.

Next, we consider filter-quotients of Θ(P ) for a sheaf P of L-structures.Recall that we already defined an L-structure P/f.

3We cannot consider a colimit lim−→U∈f a ∈M ; δ(a) = U as in the case of sheaves since

restrictions do not necessarily exist.4Notice that we use the same notations ∼f and [−]f for two different equivalence relations on

M and O(X).

21

Lemma 3.6. Let P be a sheaf of L-structures and (M, δ) := Θ(P ). Then thecanonical map P/f → M/f induces a bijection between P/f and the set ofglobal elements ofM/f.

Proof. Since δf([a]f) = [1X ]f iff δ(a) ∈ f, it is obvious that the image of thecanonical map P/f → M/f consists of global elements. We will show thismap is injective.

For a, b ∈ M with δ(a), δ(b) ∈ f, they belong to the same equivalenceclass in P/f = lim−→U∈f

P (U) if and only if there exists U ∈ f such that U ≤δ(a) ∧ δ(b) and a|U = b|U . On the other hand, by [FS79, Proposition 4.7(viii)]and separatedness,

a ∼f b ⇐⇒[δ(a) ∨ δ(b)→

∨W ≤ δ(a) ∧ δ(b) ; a|W = b|W

]∈ f,

⇐⇒ W0 :=∨

W ∈ O(X) ; a|δ(a)∧W = b|δ(b)∧W∈ f.

Again by separatedness, a|δ(a)∧W0= b|δ(b)∧W0

. Thus, the map P/f → M/f isinjective.

Note that the map P/f → M/f is not surjective even if X is a discretespace.

To give a generalization of the construction of P/f to Heyting-valued struc-tures, we need to discuss how and when an ordinary structure can be obtainedfrom some “local sections” of a Heyting-valued structure. The following con-struction is an analogue of Definition 3.1(1).

From an O(X)-valued L-structure (M, δ), we would like to construct anordinary L-structure Γ(U,M) as follows. Set Γ(U,M) := a ∈M ; δ(a) = U for U ∈ O(X). We would like to make Γ(U,M) into an L-structure so that, forany relation R and any a ∈ Γ(U,M)n,

Γ(U,M) |= R(a)def.⇐==⇒ RM(a) = U.

To define an interpretation fΓ(U,M) : Γ(U,M)n → Γ(U,M) for each functionsymbol f , we have to demand the following:

AssumptionFor each function symbol f , the morphism fM : (Mn, δ) → (M, δ) is

represented by some map h : Mn →M satisfying δ(a,a′) ≤ δ(h(a), h(a′))

and δ(h(a)) = δ(a) for any a,a′.

By the observation we made in the definition of the functor Θ at the beginningof §1.2, any Heyting-valued structure of the form Θ(P ) satisfies the Assump-tion. ForM satisfying the Assumption, we can suitably define fΓ(U,M) to bethe restriction of h to Γ(U,M) and obtain an L-structure Γ(U,M). The satis-faction relation Γ(U,M) |= ϕ(a) is defined as usual. The reader should noticethat the relations Γ(U,M) |= ϕ(a) and ‖ϕ(a)‖ = U do not coincide in general.

Given a filter f on O(X), we write Γ(M/f) for the set of global elementsof the (O(X)/f)-valued L-structureM/f. IfM satisfies the Assumption, then

22

so does M/f, and Γ(M/f) becomes an L-structure. The resulting structureΓ(M/f) will play an essential role in describing our theorems.

Returning to the case of (M, δ) = Θ(P ), we have the desired result.

Proposition 3.7. Γ(M/f) is isomorphic to the L-structure P/f under the bijec-tion in Lemma 3.6.

Proof. By the above constructions and Definition 3.1(2),

Γ(M/f) |= R([a]f)def.⇐==⇒ RM/f([a]f) := [RM(a)]f = [1X ]f,

⇐⇒ RM(a) =∨

W ≤ δ(a) ; a|W ∈ RP (W )∈ f,

⇐⇒ ∃W ∈ f, P (W ) |= R(a|W ),

⇐⇒ P/f |= R([a]f).

Thus, the construction of Γ(M/f) indeed generalizes that of P/f. In theremainder of this section, letM be an O(X)-valued L-structure satisfying theAssumption.

3.3 Łos’s Theorem

Łos-type theorems for sheaves of structures appeared in [Ell74, p. 179, Ul-trastalk Theorem] (see §3.3.1), [Bru16, Theorem 2.6 attributed to F. Miraglia],and [Cai95, Teorema 5.2]. The first two of them restrict themselves to ∀-freeformulas. Caicedo’s result is closer to ours, but no proof is given there. Wegive a generalization of Łos’s theorem improving all these results, and alsogive a characterization of Heyting-valued structures for which Łos’s theoremholds w.r.t. any maximal filter, which generalizes a similar theorem in [PV20,Theorem 2.8] for Boolean-valued structures consisting of global elements only.

Definition 3.8. For each L-formula ϕ, the Gödel translation ϕG is definedinductively:

• ⊥G ≡ ⊥, and ϕG ≡ ¬¬ϕ if ϕ is atomic but not ⊥.

• (ϕ ∧ ψ)G ≡ ϕG ∧ ψG, (ϕ ∨ ψ)G ≡ ¬(¬ϕG ∧ ¬ψG),

• (ϕ→ ψ)G ≡ ϕG → ψG,

• (∀vϕ(v,u))G ≡ ∀vϕG(v,u), (∃vϕ(v,u))G ≡ ¬∀v¬ϕG(v,u).

Definition 3.9. A filter f on O(X) isM-generic when it satisfies the following:

• for each closed LM-formula ϕ(a) with δ(a) ∈ f, either∥∥ϕG(a)

∥∥M ∈ f or∥∥¬ϕG(a)∥∥M ∈ f holds.

• for any LM-formula ϕ(v,a) with δ(a) ∈ f, if∥∥∃vϕG(v,a)

∥∥M ∈ f, thenthere exists b ∈M such that

∥∥ϕG(b,a)∥∥M ∈ f.

Theorem 3.10 (cf. [Cai95, Teorema 5.2]). If f is M-generic, then, for any L-formula ϕ(v) and a ∈Mn with δ(a) ∈ f,

Γ(M/f) |= ϕ([a]f) ⇐⇒∥∥ϕG(a)

∥∥M ∈ f.

23

Proof. Let Φ be the set of closed LM-formulas ϕ(a) with δ(a) ∈ f for whichthe above equivalence hold. We can easily see that Φ contains atomic formulasand is closed under the logical connectives ∧,∨,→. For example, to see thatϕ(a), ψ(a) ∈ Φ implies (ϕ(a)→ ψ(a)) ∈ Φ, we only have to show

δ(a) ∧(∥∥ϕG(a)

∥∥→ ∥∥ψG(a)∥∥) ∈ f ⇐⇒ either

∥∥ϕG(a)∥∥ /∈ f or

∥∥ψG(a)∥∥ ∈ f.

This follows immediately fromM-genericity.Suppose ϕ(b,a) ∈ Φ for any b with δ(b) ∈ f. Since ¬¬∃vϕG and ¬∀v¬ϕG

are intuitionistically equivalent,∥∥¬∀v¬ϕG(v,a)∥∥ ∈ f ⇐⇒

∥∥¬¬∃vϕG(v,a)∥∥ ∈ f,

⇐⇒∥∥∃vϕG(v,a)

∥∥ ∈ f,

⇐⇒ ∃b ∈M,∥∥ϕG(b,a)

∥∥ ∈ f,

⇐⇒ ∃b ∈M, δ(b) ∈ f and Γ(M/f) |= ϕ([b]f, [a]f),

⇐⇒ Γ(M/f) |= ∃vϕ(v, [a]f).

For the universal quantifier, we need a fact on Gödel translation. Since(ϕ ↔ ¬¬ϕ)G ≡ ϕG ↔ ¬¬ϕG holds and ϕ ↔ ¬¬ϕ is classically valid, ϕG ↔¬¬ϕG is intuitionistically valid by [vDal13, Theorem 6.2.8]. Therefore,∥∥∀vϕG(v,a)

∥∥ ∈ f ⇐⇒∥∥¬∀v¬¬ϕG(v,a)

∥∥ /∈ f,

⇐⇒∥∥∃v¬ϕG(v,a)

∥∥ /∈ f,

⇐⇒ ∀b ∈M, δ(b) ∈ f implies∥∥¬ϕG(b,a)

∥∥ /∈ f,

⇐⇒ ∀b ∈M, δ(b) ∈ f implies∥∥ϕG(b,a)

∥∥ ∈ f,

⇐⇒ ∀b ∈M, δ(b) ∈ f implies Γ(M/f) |= ϕ([b]f, [a]f),

⇐⇒ Γ(M/f) |= ∀vϕ(v, [a]f).

We say a formula is ∀-free if it is built up without ∀.

Corollary 3.11. In the above notations, suppose that either of the followingconditions holds:

• O(X) is a complete Boolean algebra.

• ϕ is ∀-free. (In particular, ϕG and ¬¬ϕ are intuitionistically equivalent.)

Then, for anyM-generic filter f and a ∈Mn with δ(a) ∈ f,

Γ(M/f) |= ϕ([a]f) ⇐⇒ ‖ϕ(a)‖M ∈ f.

A key to findingM-generic filters is the following proposition. For proofs,the reader is guided to refer [Mir88, Theorem 2.1] and [Cai95, Teorema 3.3].

Proposition 3.12 (Maximum Principle). If M is complete as an O(X)-valuedset, then, for any LM-formula ϕ(v,a), there exists b ∈M such that

‖ϕ(b,a)‖M ≤ ‖∃vϕ(v,a)‖M ≤ ‖¬¬ϕ(b,a)‖M in O(X).

We sayM satisfies the maximum principle if the conclusion holds.

24

In the topological case, the maximum principle means that we can find anopen set ‖ϕ(b,a)‖ dense in ‖∃vϕ(v,a)‖.

Remark 3.13. Volger [Vol76, p. 4] pointed out that the maximum principle forBoolean-valued structures holds under a weaker assumption:

for any aii∈I ⊆M and any (strong) anti-chain Uii∈I ⊆ O(X)

(i.e., a pairwise disjoint family) satisfying Ui ≤ δ(ai) for each i ∈ I ,there exists a ∈M such that Ui ≤ δ(a, ai) for each i ∈ I .

For detailed proof, see [PV20, Proposition 2.11], where the authors call thisthe mixing property. This does not assume any existence of restrictions ofelements, and we would like to remove such an assumption from the previ-ous proposition. However, we cannot apply their argument to Heyting-valuedstructures because the anti-chain they consider may not cover ‖∃vϕ‖ in gen-eral. Bell [Bel14] assumes that the frame in consideration is refinable to ensureexistence of an anti-chain refining ‖∃vϕ‖ and to show that a specific Heyting-valued structure satisfies the maximum principle (he calls it the ExistencePrinciple). We do not know whether the existence of restrictions and refine-ments can be removed from the previous proposition.

We also remark that all the results mentioned above on the maximumprinciple involve the use of the axiom of choice or its equivalents.

Theorem 3.14 (Main Theorem).For any O(X)-valued L-structureM satisfying the Assumption, TFAE:

(i) M satisfies the following variant of the maximum principle: for anyLM-formula ϕ(v,a), there are finitely many b1, . . . , br ∈M such that∨

i

∥∥ϕG(bi,a)∥∥M ≤ ∥∥∃vϕG(v,a)

∥∥M ≤ ¬¬∨i

∥∥ϕG(bi,a)∥∥M .

(ii) Every maximal filter on O(X) isM-generic.

(iii) For any maximal filter m on O(X) and any closed LM-formula ϕ(a)

with δ(a) ∈ m,

Γ(M/m) |= ϕ([a]m) ⇐⇒∥∥ϕG(a)

∥∥M ∈ m.

Proof. (i)⇒(ii): let m be a maximal filter on O(X). For any U ∈ O(X), eitherU ∈ m or ¬U ∈ m holds. Moreover, if U ∨ V ∈ m, then U ∈ m or V ∈ m. Thus,the maximum principle impliesM-genericity of m.(ii)⇒(iii): by Theorem 3.10.(iii)⇒(i): the following argument is a modification of the proof of [PV20, The-orem 2.8]. To simplify notations, we may assume δ(a) = 1X and suppress theparameter a. For an arbitrary a, we may use the frame O(δ(a)) = (δ(a))↓instead of O(X) in the following.

For any LM-formula ϕ(v) with∥∥∃vϕG(v)

∥∥ 6= 0X , we can take a maximalfilter m 3

∥∥∃vϕG(v)∥∥. Since ∃vϕG → ¬∀v¬ϕG is intuitionistically valid, we

have∥∥(∃vϕ(v))G

∥∥ ∈ m. By the assumption, Γ(M/m) |= ∃vϕ(v). Then thereexists b ∈ M such that δ(b) ∈ m and Γ(M/m) |= ϕ([b]m). Again by theassumption, there exists b ∈M such that

∥∥ϕG(b)∥∥ ∈ m.

25

We have just shown that any maximal filter containing∥∥(∃vϕ(v))G

∥∥ alsocontains some

∥∥ϕG(b)∥∥. Notice that

∥∥ϕG(b)∥∥ is a regular element of O(δ(b))

because ϕG ↔ ¬¬ϕG is intuitionistically valid. We write Reg(O(X)) for thecomplete Boolean algebra of regular elements of O(X). Now we consider thespectrum Spec(Reg(O(X))) of Reg(O(X)), i.e., the Stone space of ultrafilterson Reg(O(X)) whose basic (closed) open sets are of the form

D(U) := u ∈ Spec(Reg(O(X))) ; u 3 U for U ∈ Reg(O(X)).

Since maximal filters on O(X) correspond to ultrafilters on Reg(O(X)) (see[Joh82, Exercise II.4.9], [Sip, Theorem 1.44]), the above observation yields 5

D(∥∥(∃vϕ(v))G

∥∥) ⊆ ⋃b∈M

D(¬¬∥∥ϕG(b)

∥∥) .By compactness of D(

∥∥(∃vϕ(v))G∥∥), we can find b1, . . . , br such that

D(∥∥(∃vϕ(v))G

∥∥) ⊆⋃i

D(¬¬∥∥ϕG(bi)

∥∥) = D(¬¬∨i

∥∥ϕG(bi)∥∥) .

Hence, we have∥∥∃vϕG(v)

∥∥ ≤ ∥∥(∃vϕ(v))G∥∥ ≤ ¬¬∨i ∥∥ϕG(bi)

∥∥.

Combining the results in this section, we obtain

Corollary 3.15 (Classical Łos’s theorem). Let X be a set, Mxx∈X an X-indexed family of non-empty L-structures, and u an ultrafilter over X . Then,for any L-formula ϕ(u1, . . . , un) and a1, . . . , an ∈

∏xMx,∏

x

Mx/u |= ϕ([a1]u, . . . , [an]u)

⇐⇒x ∈ X ;Mx |= ϕ(a1

x, . . . , anx)∈ u.

Proof. Let P be the sheaf corresponding to the local homeomorphism∐x∈XMx →

X as in Example 3.2. The statement follows from the facts∏xMx/u ' P/u '

Γ(Θ(P )/u) and∥∥ϕ(a1, . . . , an)

∥∥Θ(P )=x ∈ X ;Mx |= ϕ(a1

x, . . . , anx)

.

We remark that Pierobon & Viale [PV20] give set-theoretic examples of

• a Boolean-valued structure which is not a sheaf but satisfies the maxi-mum principle, and

• a Boolean-valued structure violating Łos’s theorem (and the maximumprinciple).

3.3.1 Ellerman’s Viewpoint

Various Łos-type theorems for specific sheaves of structures have been con-sidered in the literature. Some of them are special cases of our theorem, but

5While∥∥ϕG(b)

∥∥ is regular in O(δ(b)), it is not necessarily regular in O(X). This is why weuse ¬¬

∥∥ϕG(b)∥∥ here.

26

others are not. For simplicity, we treat ∀-free formulas only. Let X be a topo-logical space and Spec(X) be the space of prime filters on the frame O(X).Spec(X) has the basic open set DU = p ; U ∈ p for each U ∈ O(X). Wehave a continuous map η : X → Spec(X) sending x to nx. For any sheafP of L-structures, the direct image sheaf η∗P on Spec(X) is again a sheafof L-structures. Ellerman [Ell74, p. 179] showed the following (cf. [Mul77]and [Sip]):

Theorem 3.16 (Ultrastalk Theorem). For any maximal filter m 3 U and anyclosed LΘ(P )-formula ϕ(a) with a ∈ P (U)n = (η∗P )(DU )n,

(η∗P )m |= ϕ([a]m) ⇐⇒ ‖ϕ(a)‖ ∈ m.

Our theorem subsumes the Ultrastalk Theorem since

(η∗P )m = lim−→D3m

(η∗P )(D) ' lim−→DU3m

(η∗P )(DU ) = lim−→U∈m

P (U) = P/m.

Especially, Łos theorem for unbounded Boolean ultrapowers [Man71, The-orem 1.5] is under our scope (cf. [Macn77]). However, Ellerman’s approachsuggests a significant viewpoint missing in ours: various model-theoretic con-structions are realized by taking stalks of sheaves on the spectrum of a distributivelattice. For example, as we saw in Example 3.5, a bounded Boolean ultrapoweris a stalk over an ultrafilter on a (possibly non-complete) Boolean algebra.There are Łos-type theorems for such structures, e.g., [BW79, Lemma 7.1] fora family of Boolean products. The relationship between these theorems andour approach should be explored elsewhere (see the comments in the nextsection).

4 Related Topics and Future Directions

Finally, we give an overview of various sheaf-theoretic methods in model the-ory with an expanded list of previous works, and indicate future directionsfrom a topos-theoretic perspective.

Forcing and Generic Models: We again assume all formulas are ∀-free. LetP be a sheaf of L-structures on a topological space X . As we noticed in §2.3,forcing values give the sheaf semantics in Sh(X). We can consider anotherforcing relation, for x ∈ X ,

x P ϕ(a)def.⇐==⇒ x ∈ ‖ϕ(a)‖ .

Caicedo [Cai95] called “U ” the local semantics and “x ” the punctualsemantics.

On the other hand, each stalk Px is an L-structure, and we can also con-sider the relation Px |= ϕ(ax) for each closed LM-formula ϕ(a) with x ∈ δ(a).Define the discrete value of a formula:

|ϕ(a)| := x ∈ δ(a) ; Px |= ϕ(ax) .

27

For any atomic relation R, by definition,

Px |= R(ax) ⇐⇒ ∃V 3 x, P (V ) |= R(a|V ),

i.e., |R(a)| = ‖R(a)‖. However, in general, |ϕ(a)| 6= ‖ϕ(a)‖. Some authorsconsidered the relationship between them ( [Man77, §1] and [Lou79, Theorem4.3, Lemma 5.1]).

Kaiser [Kai77] addressed the problem when the relations Px |= ϕ(ax) andx P ϕ(a) coincide for any formula. He called such Px a generic stalk. If thefilter nx is Θ(P )-generic, then Px is a generic stalk by our Łos-type theorem.Kaiser used generic stalks to obtain omitting types and consistency resultssimilar to those in [Kei73] (cf. [Cai95, §6], [BM04]).

From a topos-theoretic perspective, Blass & Scedrov [BS83] constructed theclassifying toposes of existentially closed models and finite-generic models.Their work was apparently inspired by Keisler’s viewpoint [Kei73] and mightbe related to ours.

Stalks, Global Sections, and Induced Geometric Morphisms: In additionto stalks of sheaves, the structure Γ(X,P ) of global sections is of our futureinterest (see below). The Feferman–Vaught theorem works for global sectionsjust like Łos’s theorem does for stalks. Comer [Com74] gave a sheaf-theoreticinterpretation of the original Feferman–Vaught theorem [FV59]. Feferman–Vaught type theorems and their applications to sentences preserved undertaking global sections were pursued in [Vol76], [LL85], [Man77], [Tak80] and[BW79] (cf. [Vol79]).

From a topos-theoretic viewpoint, taking stalks and global sections canbe seen as part of geometric morphisms. Any morphism f : X → Y of lo-cales (Definition 1.3) or of topological spaces induces a geometric morphism(f∗, f∗) : Sh(X)→ Sh(Y ). Then,

• The stalk Px is f∗P for the geometric morphism Set → Sh(X) inducedby the point f = x : 1→ X .

• The set P (X) is f∗P for the (essentially unique) geometric morphismSh(X)→ Set induced by f : X → 1.

Furthermore, we can construct a geometric morphism Set(O(X))→ Set(O(Y )),and it is canonically identified with (f∗, f∗) via the equivalence in Corollary1.21. Therefore, we may investigate stalks and global sections in the moregeneral framework of base change of Heyting-valued structures. This categor-ical approach has an advantage over the set-theoretic approach of [ACM19] tobase change of Heyting-valued universes since the construction of geometricmorphisms is much simpler and the logical behavior under base change alongthem is well-understood for various classes of morphisms of locales [Elephant,Chapter C3].

Sheaf Representation and Model Theory for Sheaves: Algebraic structuresoften have representations as global sections of sheaves of structures. Knoebel’s

28

monograph [Kno12] includes a brief description of a history of sheaf repre-sentations of algebras (see also [Joh82, Chapter V]). Sheaf representations overStone spaces, e.g., Pierce representation of commutative rings [Pie67], play aspecial role in model theory. Following Lipshitz & Saracino [LS73], Macin-tyre [Maci73] established a general method for obtaining model-companionsof theories whose models have sheaf representations over Stone spaces withgood stalks (cf. [Cars73]). He exploited Comer’s version of the Feferman–Vaught theorem to transfer model-theoretic properties of stalks to global sec-tions. This line of research was followed by [Wei75], [vdDri77], [Com76]and [BW79] (see also [Maci77, §6]). Later, Bunge & Reyes [BR81] gave a topos-theoretic unification (cf. [Bun81]).

In this line of research, sheaves having good stalks are often sheaf modelsof well-behaved theories. For example, any (commutative) von Neumann reg-ular ring R is represented by a sheaf of rings over a Stone space X(R) whosestalks are fields, and such a sheaf is a model of the theory of fields in thetopos Sh(X(R)). The theory of von Neumann regular rings has the model-completion, whose models are represented by “algebraically closed fields” insheaf toposes over Stone spaces. Thus, we may expect that developing modeltheory for sheaves will deepen our understanding of ordinary model theory.Model theory for sheaves has been studied intermittently by some authors.The pioneering work is [Lou79], where Loullis had already pointed out theimportance of the viewpoint we just mentioned. Our standpoint emphasizingHeyting-valued structures was greatly influenced by him too. Some other au-thors considered model-theoretic phenomena for models in various toposes( [Bel81], [Zaw83], [GV85], [Mir88], [Ack14]).

In fact, model theory for sheaves is part of what should be called topos-internal model theory or model theory in toposes. Topos-internal modeltheory concerns theories internal to toposes, and internal theories in a sheaftopos admit sheaves of function symbols and relation symbols (cf. [Hen13]). It mustbe closer to doing model theory in a Heyting-valued universe (cf. [KK99]). Theapproach by Brunner & Miraglia [Bru16], admitting a presheaf of constantsymbols in place of a set of constants, is regarded as a restricted form oftopos-internal model theory. In contrast to the scarcity of research on topos-internal model theory, there is much more on universal algebra in toposes andsheaf models for constructive mathematics.

Finally, we would like to mention a potential application of topos-internalmodel theory to algebraic geometry. At the end of [Lou79], Loullis suggeststhat algebraic geometry over von Neumann regular rings [SW75] could beobtained by doing algebraic geometry in some topos. The works of Bunge[Bun82] and her student MacCaull [MC88] reflect that idea, but no one fol-lowed them. We leave that direction as the ultimate goal of our research.

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[Ack14] N. L. Ackerman. “On Transferring Model Theoretic Theorems ofL∞,ω in the Category of Sets to a Fixed Grothendieck Topos”.

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[ACM19] J. G. Alvim, A. F. S. Cahali, and H. L. Mariano. “Induced Mor-phisms between Heyting-valued Models”. Dec. 3, 2019. arXiv:1910.08193 [math.CT].

[Bel81] J. L. Bell. “Isomorphism of Structures in S-Toposes”. In: J. Symb.Log. 46.3 (1981), pp. 449–459. doi: 10.2307/2273748.

[Bel05] J. L. Bell. Set Theory: Boolean-Valued Models and IndependenceProofs. 3rd ed. Oxford Logic Guides 47. Oxford University Press,2005.

[Bel14] J. L. Bell. Intuitionistic Set Theory. Stud. Log. (Lond.) 50. CollegePublications, 2014. Available at http://publish.uwo.ca/jbell/INTBOOK(Repaired).pdf.

[BS83] A. Blass and A. Scedrov. “Classifying Topoi and Finite Forcing”.In: J. Pure Appl. Algebra 28.2 (1983), pp. 111–140. doi: 10.1016/0022-4049(83)90085-3.

[HoCA3] F. Borceux. Handbook of Categorical Algebra. Vol. 3: Categoriesof Sheaves. Encyclopedia Math. Appl. 53. Cambridge UniversityPress, 1994.

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