Sheaves of Structures, Heyting-Valued Structures, and a Generalization of Ło´S’Stheorem

Sheaves of Structures, Heyting-Valued Structures, and a Generalization of Ło´s’sTheorem

Hisashi Aratake*

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

Abstract Sheaves of structures are useful to give constructions in universal algebra and model theory. We can describe their logical behavior in terms of Heyting-valued structures. In this paper, we ﬁrst provide a systematic treatment of sheaves of structures and Heyting-valued structures from the viewpoint of categorical logic. We then prove a form of Ło´s’stheorem for Heyting-valued structures. We also give a characterization of Heyting-valued structures for which Ło´s’stheorem holds with respect to any maximal ﬁlter.

0 Introduction

Sheaf-theoretic constructions have been used in universal algebra and model theory. In this context, sheaves of abelian groups or rings in geometry are generalized to sheaves of structures. We can obtain, for example, the product (resp. an ultraproduct) of a family of structures from some sheaf by taking the set of global sections (resp. a stalk). This viewpoint originated from the early literature [Com74], [Ell74] and [Maci73]. In combination with the theory of sheaf representations of algebras, Macintyre [Maci73] succeeded in giving model-companions of some theories of commutative rings by transferring model-theoretic properties from stalks to global sections. On the other hand, sheaves have another description as Heyting-valued sets. The notion of Heyting-valued sets originally arises from that of Boolean- valued models of set theory, which was introduced in relation to Cohen’s forcing. The development of topos theory in the early seventies, mainly due arXiv:2012.04317v1 [math.LO] 8 Dec 2020 to Lawvere & Tierney, revealed profound relationships between toposes and models of set theory; objects in a topos can be regarded as “generalized sets” in a universe. Subsequently, Fourman & Scott [FS79] and Higgs 1 independently 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 category Set(O(X)) of O(X)-valued sets turned out to be categorically equivalent.

*E-mail address: [email protected]

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

1 Some model-theorists of that era immediately applied Heyting-valued sets to concrete problems in sheaf-theoretic model theory. However, general methods of Heyting-valued model theory have not been explored enough, though Fourman & Scott mentioned such a direction in the preamble of [FS79]. Even worse, we are not aware of any clear explanation of the relationship between sheaves of structures and Heyting-valued structures. In this paper, employ- ing well-established languages of categorical logic, we will give a gentle and coherent account of Heyting-valued semantics of first-order logic from the categorical point of view, and will apply that framework to obtain a generalization of Ło´s’stheorem for Heyting-valued structures. We also provide a characterization of Heyting-valued structures for which Ło´s’stheorem holds w.r.t. any maximal filter. Our theorems improve the works of Caicedo [Cai95] and Pierobon & Viale [PV20]. While our principal examples of Heyting-valued sets are sheaves on topological spaces, other natural examples include sheaves on the complete Boolean algebra of regular open sets and Boolean-valued sets on the measure algebra. Therefore, we will develop our theory based on any complete Heyting algebra (a.k.a. a frame or a locale) rather than a topological space. The intended audience for this paper is anyone who has interests both in model theory and in categorical logic. We assume some familiarity with topos theory and first-order categorical logic. Most categorical prerequisites are covered by [SGL]. In §2.1, we will recall some elements of first-order categorical logic. The areas related to this paper (and its sequels in the future) are diverse, including model theory, universal algebra, set theory, categorical logic, topos theory, and their applications to ordinary mathematics. The author did his best to ensure that the reader can follow the scattered literature (especially in model theory and topos theory) on each occasion during the course.

The structure of this paper: In §1, we will begin with preliminaries on sheaves and Heyting-valued sets. After we see basic properties of Heyting- valued sets and morphisms between them, we will give an outline of the equivalence of sheaves and Heyting-valued sets. We also provide some details on the topos Set(O(X)) of O(X)-valued sets. In §2, we will study structures 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 constructions, we will introduce a further generalization of ﬁlter-quotients of sheaves to Heyting-valued structures and prove Ło´s’stheorem and the characterization theorem. In §4, we will indicate some possible future directions with an expanded list of previous works.

Closing the introduction, we have to mention Loullis’ work [Lou79] on Boolean-valued model theory. The starting point of this research was try- ing to digest his work from a modern categorical viewpoint, though our work is 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

2 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 the presentation 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] and by Fourman & Scott [FS79]. In this section, we will review the construction of the category Set(O(X)) of O(X)-valued sets for a locale X, its relation to sheaves on X, and its categorical structures as a topos. Most results are covered by [Hig84], [FS79], [Elephant, §C1.3] and [HoCA3, Chapter 2]. For the 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 topos obtained from a tripos (see [HJP80] and [vOos08, Chapter 2]). The internal logic 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 distributive law: _ _ a ∧ bi = a ∧ bi. i i In particular, any frame has 0 and 1. A frame is the same thing as a complete Heyting algebra: the infinitary distributive law for a frame H says that each monotone map a ∧ (−): H → H preserves arbitrary joins. This happens exactly when each map a ∧ (−) has a right 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 complete Heyting algebras (and even those for complete lattices):

Deﬁnition 1.2. Let H,H0 be frames. A frame homomorphism h: H → H0 is a map from H to H0 preserving ﬁnite meets and arbitrary joins. Let Frm denote the category of frames.

3 Similarly to the above, any frame homomorphism h: H → H0 has a right adjoint k : H0 → H given by _ k(b) = { a ∈ H ; h(a) ≤ b } .

Any continuous map f : X → Y of topological spaces gives rise to a frame homomorphism f ∗ : O(Y ) → O(X) given by f ∗(V ) = f −1(V ), where O(X) (resp. O(Y )) is the frame of open sets of X (resp. Y ). The functor O(−): Topop → Frm is full and faithful on sober spaces (i.e., spaces satisfying a suitable axiom between T0 and T2). Therefore, it translates the language of 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 ) for the 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 adjoint pt: 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-free topology, see [Joh82] and [PP12]. We are now ready to define Heyting-valued sets. In the remainder of this section, we fix a locale X. Definition 1.4. An O(X)-valued set (A, α) is a pair of a set A and a map α: 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 relation” on A. Instead of α(a, b), the notation a = b is frequently used in the literature. We introduce a few conventions: J K • 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” (again in the logic of tripos). Deﬁnition 1.5. Let (A, α), (B, β) be O(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, a0 ∈ A, ∀b, b0 ∈ B, α(a, a0) ∧ ϕ(a, b) ∧ β(b, b0) ≤ ϕ(a0, b0), ∀a ∈ A, ∀b, b0 ∈ B, ϕ(a, b) ∧ ϕ(a, b0) ≤ β(b, b0), _ ∀a ∈ A, α(a) = ϕ(a, b). b∈B

4 In particular, ϕ(a, b) ≤ α(a) ∧ β(b) always holds. If ψ :(B, β) → (C, γ) is another morphism, we can deﬁne the composite ψ ◦ ϕ by _ (ψ ◦ ϕ)(a, c) = ϕ(a, b) ∧ ψ(b, c). b∈B

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) of intuitionistic 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 and examples from set theory to investigate Heyting-valued sets (cf. [PV20]). We list useful facts on Heyting-valued sets, some of which will not be used in 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) ∧ ϕ(a, b0) b0 _ _ ≤ [ϕ(a, b0) ∧ ψ(a, b0) ∧ ψ(a, b)] ≤ [ϕ(a, b0) ∧ β(b0, b)] = ϕ(a, b). b0 b0

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, a0 ∈ A, ∀b ∈ B, ϕ(a, b) ∧ ϕ(a0, b) ≤ α(a, a0).

(2) ϕ is an epimorphism if and only if _ ∀b ∈ B, β(b) = ϕ(a, b). a∈A

(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 given −1 by ϕ (b, a) = ϕ(a, b). Deﬁnition 1.9. We say that a morphism ϕ:(A, α) → (B, β) is represented by a map h: A → B when

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

Proposition 1.10.

5 (1) A morphism ϕ:(A, α) → (B, β) is represented by a map h: A → B if and 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) if and only if ∀a, a0 ∈ A, α(a, a0) ≤ β(ha, ha0). Moreover, if h further satisﬁes α(a) = β(ha) for all a ∈ A, then the morphism 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 represent 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 another morphism. If ϕ (resp. ψ) is represented by a map h (resp. k), then ψϕ is represented by kh. Proposition 1.11. Let ϕ:(A, α) → (B, β) be a morphism represented by h. • ϕ is monic ⇐⇒ ∀a, a0 ∈ A, α(a, a0) = α(a) ∧ α(a0) ∧ β(ha, ha0). W • 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, a0 ∈ A, α(a, a0) = β(ha, ha0). W • 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 an isomorphism exactly when the following conditions hold _ ∀a, a0 ∈ A, α(a, a0) = β(ha, ha0), and ∀b ∈ B, β(b) = β(ha, b). a If h satisﬁes these conditions, the induced isomorphism ϕ(a, b) = β(ha, b) has −1 the inverse ϕ (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 the relationship between sheaves on X and O(X)-valued sets. Our presentation style 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 and U ∈ O(X), elements of PU (resp. of P 1X ) are called sections of P on U (resp. global sections of P ). If a ∈ PU and W ≤ U, we will write a|W for P (W ≤ U)(a).

6 (2) A presheaf P is said to be a sheaf on X when it satisﬁes the following W condition: For any covering {Ui}i of U ∈ O(X) (i.e., U = i Ui) and any

family {ai}i of sections ai ∈ PUi, if ai|Ui∧Uj = aj|Ui∧Uj for all i, j, then

there exists a unique a ∈ PU such that a|Ui = ai for all i. (3) Morphisms of presheaves are deﬁned to be natural transformations. op The functor category SetO(X) is also called the category of presheaves. Let Sh(X) denote its full subcategory spanned by sheaves. ` We can associate a presheaf P with an O(X)-valued set Θ(P ) := ( U P U, 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 ) are exactly global sections of P .

• If P is a sheaf, then δP (a, b) is the largest element on which the restrictions 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 on X, the induced map h: U PU → ` U QU satisﬁes

2 a ∀(a, b) ∈ PU , δP (a, b) ≤ δQ(ha, hb) and δP (a) = δQ(ha). U

Therefore, by Proposition 1.10, h represents a morphism Θ(ξ): Θ(P ) → Θ(Q). op This construction gives a functor Θ: SetO(X) → 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-valued sets are separated if the latter condition holds. To give a similar characterization 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 and α(b) = U. (If v is a partial order, such b is uniquely determined and denoted by a|U .)

7 • If a family {ai}i 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 supremum is called an amalgamation of {ai}i.)

Let CSet(O(X)) denote the full subcategory of Set(O(X)) spanned by complete O(X)-valued sets. Proposition 1.15. A presheaf P on X is a sheaf if and only if Θ(P ) is complete as 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 a function σ : A → O(X) such that

∀a, a0 ∈ A, σ(a) ∧ α(a, a0) ≤ σ(a0) and σ(a) ∧ σ(a0) ≤ α(a, a0).

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(a0) = α(a, a0), then 0 σa = σa0 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 . If W a family {ai}i 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, a0) ≤ β(ha, ha0) and α(a) = β(ha) for all a, a0 ∈ A.

Proof. For any fixed a ∈ A, the map ϕ(a, −) is a singleton of (B, β). By completeness, we can find a unique ha ∈ B such that ϕ(a, b) = β(ha, b) for every b ∈ B. This defines a map h: A → B representing ϕ and having the desired properties. This lemma and Proposition 1.10(4) yield

8 Proposition 1.19. Θ induces a categorical equivalence between Sh(X) and CSet(O(X)). On the other hand, we can also show that Set(O(X)) and CSet(O(X)) are categorically equivalent. Proposition 1.20. Let A˜ be the set of singletons on (A, α). Deﬁne a valuation α˜ on A˜ by _ α˜(σ, τ) := σ(a) ∧ τ(a). a∈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˜) the completion of (A, α). For a morphism ϕ:(A, α) → (B, β), let ϕ˜ be the composite of ϕ (A,˜ α˜) →∼ (A, α) −→ (B, β) →∼ (B,˜ β˜).

Then (−˜ ): Set(O(X)) → CSet(O(X)) becomes a functor and also gives a quasi-inverse of the inclusion functor. Corollary 1.21. The categories Sh(X), Set(O(X)) and CSet(O(X)) are categorically 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)). Most results 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 a terminal object in Set(O(X)).

(2) Let {(Ai, αi)}i∈I be a ﬁnite family of O(X)-valued sets. Deﬁne a val- Q 0 V 0 0 uation δ on i Ai by δ(a, a ) = i αi(ai, ai) for a = {ai}i∈I and a = 0 Q Q {ai}i∈I . Then ( i Ai, δ) equipped with the canonical projections ( i Ai, 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 a 0 0 W valuation 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 equalizer of ϕ and ψ in Set(O(X)).

ϕ ψ (4) Let (A, α) −→ (C, γ) ←− (B, β) be morphisms of O(X)-valued sets. Deﬁne a valuation δ on A × B by _ δ((a, b), (a0, b0)) = α(a, a0) ∧ β(b, b0) ∧ ϕ(a, c) ∧ ψ(b, c). c∈C Then (A × B, δ) equipped with the canonical projections is a pullback of that diagram in Set(O(X)).

9 The following notion of strict relation is crucial in handling subobjects of an O(X)-valued set. In the next section, it will enable us to deﬁne the “forcing values” 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, a0 ∈ A, σ(a) ∧ α(a, a0) ≤ σ(a0) 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) ordered by 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 subobjects of (A, α).

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

0 0 0 0 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 a monomorphism ισ :(A, ασ) (A, α) by Proposition 1.11. Conversely, for a monomorphism ϕ:(B, β) (A, α), we deﬁne a strict relation ρϕ : A → O(X) by _ ρϕ(a) := ϕ(b, a). b∈B 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 as above, and the image factorization of ϕ is given by ϕ (B, β) (A, α)

ϕ ι(ρϕ) (A, α ) (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), σi (a) = σi(a), i i (σ → τ)(a) = α(a) ∧ (σ(a) → τ(a)).

10 Proposition 1.26. Let ϕ:(B, β) → (A, α) be a morphism. Pulling back subobjects along ϕ defines a frame homomorphism ϕ∗ : P(A, α) → P(B, β) such that, for σ ∈ P(A, α), _ ^ (ϕ∗σ)(b) = ϕ(b, a) ∧ σ(a) = β(b) ∧ [ϕ(b, a) → σ(a)]. a∈A a∈A Proof. For the last identity, note that ^ _ ^ β(b) ∧ [ϕ(b, a) → σ(a)] = ϕ(b, a) ∧ [ϕ(b, a0) → σ(a0)] a∈A a∈A a0∈A _ ≤ ϕ(b, a) ∧ [ϕ(b, a) → σ(a)] a∈A _ ≤ ϕ(b, a) ∧ σ(a). a∈A ∗ Proposition 1.27. In the same notations as above, ϕ has both a left adjoint ∃ϕ and a right adjoint ∀ϕ: for τ ∈ P(B, β), _ (∃ϕτ)(a) = ϕ(b, a) ∧ τ(b), b∈B ^ (∀ϕτ)(a) = α(a) ∧ [ϕ(b, a) → τ(b)]. b∈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 morphism defined 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 to the strict relation idO(X) on (O(X), δ), the pullback of t along χ is given W by the strict relation σ(a) = U χ(a, U) ∧ U. Conversely, given a strict relation 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) and Hom((A, α), (O(X), δ)).

! (A, ασ) ({∗}, >) y ισ t χ (A, α) (O(X), δ) Similarly to O(X), P(A, α) is not only a frame but also an O(X)-valued set. Proposition 1.29. The power object of (A, α) is given by P(A, α) equipped with the valuation 2 ^ α¯(σ, τ) := σ(a) ↔ τ(a). a∈A 2Note that α¯ does not necessarily coincide with α˜ on A˜ in Proposition 1.20. Indeed, while α˜(σ, τ) ≤ α¯(σ, τ) holds for any σ, τ ∈ A˜, the converse inequality does not hold if σ = τ = σa for a 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) = ϕ(b, τ) ∧ τ(a). τ∈P(A,α) On the other hand, for any strict relation θ, we have a morphism ^ ϕ(b, τ) = β(b) ∧ θ(b, a) ↔ τ(a). a∈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) and Set(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’s account [Cara14], which is a preliminary version of the first two chapters of her book [TST]. Although many fragments of (possibly infinitary) first- order logic are considered in the context of categorical logic, we restrict our attention to single-sorted intuitionistic logic. Examples of other fragments include 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 is associated with a natural number n (the arity of f). If n = 0, f is called a constant (symbol). • A set L-Rel of relation symbols. Each relation symbol R ∈ L-Rel is associated with a natural number n (the arity of R). If n = 0, R is called an 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 is called the empty context and denoted by []. (2) We say that a context u is suitable for an L-formula ϕ when u contains all the free variables of ϕ. A formula ϕ equipped with a suitable context u 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) 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 contains no 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-structure M in E is given by specifying the following data: • We have the underlying object |M| ∈ E and denote the n-ary product by n 0 |M| . In particular, |M| is the terminal object 1E . • To an n-ary function symbol f, we assign a morphism f M : |M|n → |M|. • To an n-ary relation symbol R, we assign a subobject RM of |M|n. As usual, we will not distinguish M and its underlying object |M| in notation. Interpretations of L-terms and L-formulas are defined by using internal operations in E. Definition 2.4 (Interpretations of terms). Let M be an L-structure in a topos E. For an L-term-in-context t(u), we define the interpretation tM : Mn → M inductively. M n • If t is a variable ui, then t is the i-th product projection πi : M → 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:

htM,...,tMi M Mn −−−−−−−→M1 m m −−→Ms ,

M M M where ht1 , . . . , tm i is the morphism obtained from the morphisms ti m by using the universal property of the product M . Definition 2.5 (Interpretations of formulas). Let M be an L-structure in a topos E. For an L-formula-in-context ϕ(u), we define the interpretation u. ϕ M as a subobject of Mn inductively. (We drop the subscript M if no confusionJ K arises.) • If ϕ ≡ (s(u) = t(u)) where s, t are terms, then u. ϕ is defined to be the equalizer of J K sM Mn M tM .

• If ϕ ≡ R(t1(u), . . . , tm(u)), then u. ϕ is the pullback J K u.R(t1, . . . , tm) RM J K

htM, . . . , tMi n 1 m m M M .

13 • If ϕ ≡ >, ⊥, ψ ∧ θ, ψ ∨ θ, ψ → θ or ¬ψ, then u. ϕ is deﬁned as expected by using the Heyting operations on Sub(MJn). K

• If ϕ ≡ ∃vψ(u, v), then u. ϕ is the image as in the following diagram: J K u, v. ψ Mn × M J K π

u. ∃vψ Mn J K , where π is the projection onto Mn.

n • If ϕ ≡ ∀vψ(u, v), then u. ϕ := ∀π u, v. ψ , where ∀π : Sub(M ×M) → n J K ∗ J K Sub(M ) is the right adjoint of π . In this paper, we will not consider the notions of models of a theory in a topos nor homomorphisms between structures.

2.2 Sheaves of Structures and Heyting-Valued Structures We now investigate the relationship between structures in Sh(X) and those in Set(O(X)). We ﬁrst consider the case of sheaves on a topological space X. Let LH be the category of topological spaces and local homeomorphisms between them. Recall that the slice category LH/X is categorically equivalent to Sh(X). Comer [Com74], Ellerman [Ell74], and Macintyre [Maci73] used the following notion to obtain model-theoretic results: Deﬁnition 2.6. A sheaf of L-structures (on X) is a tuple

X, E, π, f Ex ; x ∈ X, f ∈ L-Func , REx ; x ∈ X,R ∈ L-Rel such that • π : E → X is a local homeomorphism of topological spaces,

Ex Ex • each stalk Ex equipped with {f }f and {R }R is an L-structure, and

` n – for each function symbol f, the map x(Ex) → E induced by Ex {f }x is continuous,

` Ex ` n – for each relation symbol R, the subset x R ⊆ x(Ex) is open, ` n n where x(Ex) is seen as a subspace of the product space E for n > 0, ` 0 and x(Ex) ' X. Sheaves of abelian groups or of rings in geometry are, of course, such examples for suitable languages. We will meet other examples which give model-theoretic constructions of (usual Set-valued) structures in the next section. Lemma 2.7. A sheaf of L-structures is identiﬁed with an L-structure in LH/X. Proof. Notice the following facts:

14 ` n • x(Ex) 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-saving notations. If (M, δ) is an O(X)-valued set, then the n-th power Mn is canonically equipped with the valuation as in Proposition 1.22(2). For tuples a, a0 ∈ n 0 V 0 V M , we simply write δ(a, a ) (resp. δ(a)) for i δ(ai, ai) (resp. i δ(ai)). These notations are useful, but, in the case n = 2, we will always write δ(a) ∧ δ(b) for δ((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 topos Set(O(X)), i.e., it consists of the following data: • an O(X)-valued set (M, δ), • for each function symbol f, a morphism f M :(Mn, δ) → (M, δ), • for each relation symbol R, a strict relation RM : Mn → O(X). 2 The interpretation of equality is the diagonal (M, δ) (M , δ), which corresponds to the strict relation (a, b) 7→ δ(a, b) on (M2, δ) under the bijection in Proposition 1.24. Fourman & Scott [FS79, p. 365] defined Heyting-valued structures in a slightly less general form. Structures in the topos associated with a tripos are discussed in [vOos08, p. 69]. op Recall the construction of Θ: SetO(X) → 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- n ` n morphic to Θ(P ) as O(X)-valued sets. Indeed, the canonical map h: U (PU) → ` n n ∼ n ( U PU) represents an isomorphism ι: Θ(P ) → Θ(P ) so that ι(b, a) = −1 ι (a, b) = δP (h(b), a). Moreover, for a strict relation σ on Θ(P n), the corresponding strict relation τ on Θ(P )n is given by _ τ(a) = σ(b) ∧ δP (h(b), a). b∈Θ(P n) 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 of op categories. We can also see directly that Θ: SetO(X) → Set(O(X)) preserves finite products by using Corollary 1.12. For a given σ, by the proof of Proposition 1.24, the corresponding sub- n ` n 0 0 object of Θ(P ) is ( U (PU) , (δP n )σ) with (δP n )σ(b, b ) = σ(b) ∧ δP n (b, b ). Hence, τ is given by

_ 0 0 _ τ(a) = (δP n )σ(b, b ) ∧ ι(b , a) = σ(b) ∧ δP (h(b), a). b,b0∈Θ(P n) b∈Θ(P n)

15 We remark that τ(h(b)) = σ(b) for any b ∈ Θ(P n) and therefore τ is an extension of σ along h. Proposition 2.10. If P is a sheaf of L-structures on X, then we can make the O(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-structure M. For each function f, we have a morphism f P : P n → P of sheaves. This induces a morphism Θ(f P ): Θ(P n) → Θ(P ). By the previous lemma, we obtain a morphism f M : Θ(P )n → Θ(P ), which can be computed as

M 0 _ −1 P 0 P 0 f (a, a ) = ι (a, b) ∧ Θ(f )(b, a ) = δP (f (a|δP (a)), a ) b∈Θ(P n)

n 0 for a ∈ Θ(P ) and a ∈ Θ(P ), where a|δP (a) = (a1|δP (a), . . . , an|δP (a)). In par- M n P ticular, f is represented by the map k : M → M with k(a) = f (a|δP (a)). P n For each relation R, we have a subsheaf R P . This induces a sub- P n object Θ(R ) Θ(P ), which corresponds to the following strict relation ` n n σ : U (PU) → O(X): for b ∈ (PU) ,

_ 0 _ P σ(b) = δP n (b, b ) = W ≤ U ; b|W ∈ R (W ) . b0∈Θ(RP )

By the previous lemma, we obtain a subobject of Θ(P )n, which corresponds M n to the following strict relation R : M → O(X): for a ∈ PU1 × · · · × PUn, _ RM(a) = ι−1(a, b) ∧ σ(b) b∈Θ(P n) _ h _ P i = δP (h(b), a) ∧ W ≤ δP n (b); b|W ∈ R (W ) b∈Θ(P n) _ P = W ≤ U1 ∧ · · · ∧ Un ; a|W ∈ R (W ) .

2 Notice that the subobject Θ(P ) Θ(P ) obtained from the diagonal P 2 P is the same as the one determined by the strict relation δP on Θ(P ). We could describe the converse construction (from Heyting-valued structures to sheaves of structures). This involves a complicated use of completion of Heyting-valued sets, and we do not ﬁnd such details to be useful for the purpose 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 universes to mathematics (cf. [KK99] and the references therein). From the viewpoint 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 we first define them categorically and then observe that our definition is compatible with the usual one. The categorical description seems to be folklore but 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 new constant symbol for each element of M.

Deﬁnition 2.12. For an L-formula-in-context ϕ(u), the strict relation kϕ(−)kM n on (M, δ) is deﬁned to be the one corresponding to the subobject u. ϕ (M,δ) n n M J K (M, δ) . For a ∈ M , kϕ(a)k is called the forcing value of the closed LM- formula ϕ(a). We drop the superscript M if no confusion arises. Since the strict relation a 7→ δ(a) is the greatest element in P(Mn, δ), kϕ(a)kM ≤ δ(a) always holds. Using the results in §1.3, we can calculate the forcing values inductively.

Proposition 2.13.

M _ M M M kR(t1(a), . . . , tm(a))k = ht1 , . . . , tm i(a, b) ∧ R (b), b∈Mm M _ ks(a) = t(a)k = hsM, tMi(a, (b, c)) ∧ δ(b, c), b,c∈M kϕ(a) ∧ ψ(a)kM = kϕ(a)kM ∧ kψ(a)kM , kϕ(a) ∨ ψ(a)kM = kϕ(a)kM ∨ kψ(a)kM , h i kϕ(a) → ψ(a)kM = δ(a) ∧ kϕ(a)kM → kψ(a)kM ,

M _ M k∃vϕ(a, v)k = kϕ(a, b)k , b∈M M ^ h Mi k∀vϕ(a, v)k = δ(a) ∧ δ(b) → kϕ(a, b)k . b∈M

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

kϕ(a, b)k = kϕ(a)k ∧ 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 and completeness theorems for Heyting-valued semantics are usually formulated with respect to intuitionistic predicate logic with existence predicate (for short, IQCE) as in [TvD88, §2.2, §13.6]. However, we will only need soundness of the following form:

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

17 Proof. The assumption implies u. ϕ ≤ u. ψ as subobjects of (M, δ)n. There- J K J K 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)-valued L-structure obtained from Proposition 2.10. We can see (1) For any L-term t(u), the morphism tM is represented by the map Mn 3 P P n a 7→ t (a|δ(a)) ∈ M where t : P → P is the interpretation of t by P . n (2) For any atomic L-formula R(t1(u), . . . , tm(u)) and a ∈ M , M kR(t1(a), . . . , tm(a))k M P P = R (t1 (a|δ(a)), . . . , tm(a|δ(a))) _ P P P = W ≤ δ(a);(t1 (a|W ), . . . , tm(a|W )) ∈ R (W ) . Similarly for the formula s(u) = t(u). More generally, the forcing value kϕ(−)k for Θ(P ) can be described in terms of the subsheaf u. ϕ of P n. Let Ω be the sheaf U 7→ Ω(U) = (U)↓. This is a subobject classifierJ K in Sh(X), and we thus obtain the characteristic morphism χ: P n → Ω by the universality of the subobject classifier: ! u. ϕ 1 J yK _ true χU (a) = { W ≤ U ; a|W ∈ u. ϕ (W ) } . J K χ P n Ω Using Proposition 1.28 and the fact that Θ(Ω) and (O(X), δ) in that proposition are canonically isomorphic, we can verify the following: Proposition 2.16 (definable subsheaves and forcing values). In the above no- M n P tation, χU (a) = kϕ(a)k for any a ∈ P U. We will denote χ by kϕ(−)k and P its component χU by kϕ(−)kU . op op Let y: O(X) → SetO(X) be the Yoneda embedding, and a: SetO(X) → Sh(X) the associated sheaf functor. We write a: ayU → P n for the morphism corresponding to a ∈ P nU under the bijection n n n P U ' HomSetO(X)op (yU, P ) ' HomSh(X)(ayU, P ). In terms of forcing values, the sheaf semantics in Sh(X) (cf. [SGL, §VI.7]) has a simple description:

U P ϕ(a) def. n n ⇐==⇒ the morphism a: ayU → P factors through the subsheaf u. ϕ P , J K ⇐⇒ kϕ(−)kP ◦ a = true ◦ !, P ⇐⇒ kϕ(a)kU = 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 and Ło´s’s Theorem

As we promised after Deﬁnition 2.6, we will observe that sheaves of structures give some constructions in model theory. These constructions can be generalized to constructions for Heyting-valued structures, and they provide an adequate setup to state our Ło´s-typetheorem.

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:

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

(2) For a ﬁlter f on O(X), the colimit P/f := lim P (U) is the quotient −→U∈f ` of U∈f P (U) by the following equivalence relation: for U, V ∈ f and a ∈ P (U), b ∈ P (V ),

def. (U, a) ∼ (V, b) ⇐==⇒ ∃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. Let δ be the valuation of Θ(P ). We make P/f into an L-structure as follows:

P/f P f ([a]f) := [f (a1|δ(a), . . . , an|δ(a))]f, def. P P/f |= R([a]f) ⇐==⇒ ∃W ∈ f, a|W ∈ R (W ),

⇐⇒ ∃W ∈ f,P (W ) |= R(a|W ).

In particular, if X is a topological space and x ∈ X, each stalk Px is the quotient 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 {Mx}x∈X Q of L-structures, the product N := x∈X Mx is an L-structure such that, for i i any elements a = {ax}x∈X ,

N 1 n Mx 1 n f (a , . . . , a ) := f (ax, . . . , ax) x∈X , 1 n def. 1 n N |= R(a , . . . , a ) ⇐==⇒ ∀x ∈ X, Mx |= R(ax, . . . , ax).

Giving an X-indexed family of L-structures is the same as giving a sheaf of L-structures on the discrete space X. Let P be the sheaf corresponding to ` the local homeomorphism x∈X Mx → 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,

1 n Θ(P ) 1 n ϕ(a , . . . , a ) = x ∈ X ; Mx |= ϕ(ax, . . . , ax)

1 n holds for any formula ϕ and a , . . . , a ∈ N .

19 Example 3.3 (Ultraproducts). Let u be an ultraﬁlter over a set X. In the same Q notation as the previous example, the ultraproduct x Mx/u is the quotient Q of x Mx by the equivalence relation def. a ∼ b ⇐==⇒ { x ∈ X ; ax = bx } ∈ u equipped with canonical interpretations of L, e.g.,

Y 1 n def. 1 n Mx/u |= R([a ]u,..., [a ]u) ⇐==⇒ x ∈ X ; Mx |= R(ax, . . . , ax) ∈ u. x Q If each Mx is non-empty, x Mx/u can be described as a ﬁlter-quotient of the ` Q sheaf P corresponding to x∈X Mx → X. Since P (U) = x∈U Mx and each local section can be extended to a global section by non-emptiness, we have Y u u Mx/ ' −→lim P (U) = P/ . x U∈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 to global 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 M be an L-structure. We then have the sheaf P on the Stone space X dual to B determined 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 the bounded Boolean power of M (cf. [Hod93, §9.7]). Example 3.5 (Bounded Boolean Ultrapowers). In the same notation as the previous example, for any s, t ∈ M[B]ω, the subsets

kR(s1, . . . , sn)k = { v ∈ X ; M |= R(s1(v), . . . , sn(v)) } , ks = tk = { v ∈ X ; s(v) = t(v) } are clopen and identiﬁed with elements of B. Let u be an ultraﬁlter on B (= a point of X). The bounded Boolean ultrapower M[B]ω/u is given by

s ∼ t ⇐def==⇒. ks = tk ∈ u,

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

M[B]ω/u has a representation as a ﬁlter-quotient u M[B]ω/ ' −→lim P (DU ) ' Pu, U∈u 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 they involve more complicated sheaf-theoretic constructions. Fish [Fis00] gives a survey of bounded and unbounded Boolean (ultra)powers. These constructions 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. We use 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-structure M/f is defined as follows: 3 we first observe

Claim. The following relation ∼f on M is an equivalence relation

def. a ∼f b ⇐==⇒ [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 quotient M/∼f by M/f and the equivalence class of a ∈ M by [a]f. In particular, by applying this to the O(X)-valued set (O(X), ∧), for which 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 make M/f into an (O(X)/f)-valued set except that O(X)/f is not necessarily 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], but such a complication will not be necessary for this paper because we will use M/f forcing values kϕ([a]f)k only for atomic formulas ϕ. M/f n For each function f and each relation R, the morphism f : ((M/f) , δf) → M/f n (M/f, δf) and the strict relation R :(M/f) → O(X)/f are defined canonically: M/f M f ([a1]f,..., [an]f, [b]f) := [f (a1, . . . , an, b)]f, M/f M R ([a1]f,..., [an]f) := [R (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 of M 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 { a ∈ M ; δ(a) = U } as in the case of sheaves since −→U∈f restrictions do not necessarily exist. 4 Notice 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 the canonical map P/f → M/f induces a bijection between P/f and the set of global elements of M/f.

Proof. Since δf([a]f) = [1X ]f iff δ(a) ∈ f, it is obvious that the image of the canonical map P/f → M/f consists of global elements. We will show this map is injective. For a, b ∈ M with δ(a), δ(b) ∈ f, they belong to the same equivalence class in P/f = lim P (U) if and only if there exists U ∈ f such that U ≤ −→U∈f δ(a) ∧ δ(b) and a|U = b|U . On the other hand, by [FS79, Proposition 4.7(viii)] and separatedness, h _ i 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 is injective. Note that the map P/f → M/f is not surjective even if X is a discrete space. To give a generalization of the construction of P/f to Heyting-valued structures, we need to discuss how and when an ordinary structure can be obtained from some “local sections” of a Heyting-valued structure. The following construction is an analogue of Deﬁnition 3.1(1). From an O(X)-valued L-structure (M, δ), we would like to construct an ordinary 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, for any relation R and any a ∈ Γ(U, M)n,

Γ(U, M) |= R(a) ⇐def==⇒. RM(a) = U.

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

Assumption For each function symbol f, the morphism f M :(Mn, δ) → (M, δ) is represented by some map h: Mn → M satisfying δ(a, a0) ≤ δ(h(a), h(a0)) and δ(h(a)) = δ(a) for any a, a0.

By the observation we made in the definition of the functor Θ at the beginning of §1.2, any Heyting-valued structure of the form Θ(P ) satisfies the Assump- tion. For M satisfying the Assumption, we can suitably define f Γ(U,M) to be the 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 notice that the relations Γ(U, M) |= ϕ(a) and kϕ(a)k = U do not coincide in general. Given a filter f on O(X), we write Γ(M/f) for the set of global elements of the (O(X)/f)-valued L-structure M/f. If M satisfies the Assumption, then

22 so does M/f, and Γ(M/f) becomes an L-structure. The resulting structure Γ(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 bijection in Lemma 3.6. Proof. By the above constructions and Deﬁnition 3.1(2),

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

M _ P ⇐⇒ R (a) = W ≤ δ(a); a|W ∈ R (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 the remainder of this section, let M be an O(X)-valued L-structure satisfying the Assumption.

3.3 Ło´s’s Theorem Ło´s-typetheorems 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 ∀-free formulas. Caicedo’s result is closer to ours, but no proof is given there. We give a generalization of Ło´s’stheorem improving all these results, and also give a characterization of Heyting-valued structures for which Ło´s’stheorem holds 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 defined inductively: • ⊥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,

G G G G • (∀vϕ(v, u)) ≡ ∀vϕ (v, u), (∃vϕ(v, u)) ≡ ¬∀v¬ϕ (v, u). Definition 3.9. A filter f on O(X) is M-generic when it satisfies the following:

G M • for each closed LM-formula ϕ(a) with δ(a) ∈ f, either ϕ (a) ∈ f or M ¬ϕG(a) ∈ f holds.

G M • for any LM-formula ϕ(v, a) with δ(a) ∈ f, if ∃vϕ (v, a) ∈ f, then G M there exists b ∈ M such that ϕ (b, a) ∈ 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,

G M Γ(M/f) |= ϕ([a]f) ⇐⇒ ϕ (a) ∈ f.

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

G G G G δ(a) ∧ ϕ (a) → ψ (a) ∈ f ⇐⇒ either ϕ (a) ∈/ f or ψ (a) ∈ f.

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

G G ¬∀v¬ϕ (v, a) ∈ f ⇐⇒ ¬¬∃vϕ (v, a) ∈ f, G ⇐⇒ ∃vϕ (v, a) ∈ f, G ⇐⇒ ∃b ∈ M, ϕ (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,

G G ∀vϕ (v, a) ∈ f ⇐⇒ ¬∀v¬¬ϕ (v, a) ∈/ f, G ⇐⇒ ∃v¬ϕ (v, a) ∈/ f, G ⇐⇒ ∀b ∈ M, δ(b) ∈ f implies ¬ϕ (b, a) ∈/ f, G ⇐⇒ ∀b ∈ M, δ(b) ∈ f implies ϕ (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 following conditions holds: • O(X) is a complete Boolean algebra.

• ϕ is ∀-free. (In particular, ϕG and ¬¬ϕ are intuitionistically equivalent.) Then, for any M-generic ﬁlter f and a ∈ Mn with δ(a) ∈ f,

M Γ(M/f) |= ϕ([a]f) ⇐⇒ kϕ(a)k ∈ f. A key to ﬁnding M-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)-valued set, then, for any LM-formula ϕ(v, a), there exists b ∈ M such that

kϕ(b, a)kM ≤ k∃vϕ(v, a)kM ≤ k¬¬ϕ(b, a)kM in O(X).

We say M satisﬁes the maximum principle if the conclusion holds.

24 In the topological case, the maximum principle means that we can find an open set kϕ(b, a)k dense in k∃vϕ(v, a)k. Remark 3.13. Volger [Vol76, p. 4] pointed out that the maximum principle for Boolean-valued structures holds under a weaker assumption: for any {ai}i∈I ⊆ M and any (strong) anti-chain {Ui}i∈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 this the mixing property. This does not assume any existence of restrictions of elements, and we would like to remove such an assumption from the previous proposition. However, we cannot apply their argument to Heyting-valued structures because the anti-chain they consider may not cover k∃vϕk in general. Bell [Bel14] assumes that the frame in consideration is refinable to ensure existence of an anti-chain refining k∃vϕk and to show that a specific Heyting- valued structure satisfies the maximum principle (he calls it the Existence Principle). 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 maximum principle involve the use of the axiom of choice or its equivalents. Theorem 3.14 (Main Theorem). For any O(X)-valued L-structure M satisfying the Assumption, TFAE: (i) M satisfies the following variant of the maximum principle: for any LM-formula ϕ(v, a), there are finitely many b1, . . . , br ∈ M such that

_ G M G M _ G M ϕ (bi, a) ≤ ∃vϕ (v, a) ≤ ¬¬ ϕ (bi, a) . i i

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

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

G M Γ(M/m) |= ϕ([a]m) ⇐⇒ ϕ (a) ∈ m.

Proof. (i)⇒(ii): let m be a maximal ﬁlter on O(X). For any U ∈ O(X), either U ∈ m or ¬U ∈ m holds. Moreover, if U ∨ V ∈ m, then U ∈ m or V ∈ m. Thus, the maximum principle implies M-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 the parameter a. For an arbitrary a, we may use the frame O(δ(a)) = (δ(a))↓ instead of O(X) in the following. G For any LM-formula ϕ(v) with ∃vϕ (v) 6= 0X , we can take a maximal

ﬁlter 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 there exists b ∈ M such that δ(b) ∈ m and Γ(M/m) |= ϕ([b]m). Again by the assumption, there exists b ∈ M such that ϕG(b) ∈ m.

We have just shown that any maximal ﬁlter containing (∃vϕ(v))G also contains 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 the complete Boolean algebra of regular elements of O(X). Now we consider the spectrum Spec(Reg(O(X))) of Reg(O(X)), i.e., the Stone space of ultraﬁlters on 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

G [ G D (∃vϕ(v)) ⊆ D ¬¬ ϕ (b) . b∈M

G By compactness of D( (∃vϕ(v)) ), we can ﬁnd b1, . . . , br such that

G [ G W G D (∃vϕ(v)) ⊆ D ¬¬ ϕ (bi) = D ¬¬ i ϕ (bi) . i

G G W G Hence, we have ∃vϕ (v) ≤ (∃vϕ(v)) ≤ ¬¬ i ϕ (bi) . Combining the results in this section, we obtain

Corollary 3.15 (Classical Ło´s’s theorem). Let X be a set, {Mx}x∈X an X- indexed family of non-empty L-structures, and u an ultraﬁlter over X. Then, 1 n Q for any L-formula ϕ(u1, . . . , un) and a , . . . , a ∈ x Mx,

Y 1 n Mx/u |= ϕ([a ]u,..., [a ]u) x 1 n ⇐⇒ x ∈ X ; Mx |= ϕ(ax, . . . , ax) ∈ u. ` Proof. Let P be the sheaf corresponding to the local homeomorphism x∈X Mx → Q X as in Example 3.2. The statement follows from the facts x Mx/u ' P/u ' 1 n Θ(P ) 1 n Γ(Θ(P )/u) and ϕ(a , . . . , a ) = x ∈ X ; Mx |= ϕ(ax, . . . , ax) . We remark that Pierobon & Viale [PV20] give set-theoretic examples of

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

• a Boolean-valued structure violating Ło´s’stheorem (and the maximum principle).

3.3.1 Ellerman’s Viewpoint Various Ło´s-typetheorems for speciﬁc sheaves of structures have been considered 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 we use ¬¬ ϕG(b) here.

26 others are not. For simplicity, we treat ∀-free formulas only. Let X be a topological 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). We have a continuous map η : X → Spec(X) sending x to nx. For any sheaf P of L-structures, the direct image sheaf η∗P on Spec(X) is again a sheaf of 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 any n n closed LΘ(P )-formula ϕ(a) with a ∈ P (U) = (η∗P )(DU ) ,

(η∗P )m |= ϕ([a]m) ⇐⇒ kϕ(a)k ∈ m. Our theorem subsumes the Ultrastalk Theorem since

m (η∗P )m = lim−→ (η∗P )(D) ' −→lim (η∗P )(DU ) = lim−→ P (U) = P/ . D3m DU 3m U∈m

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

4 Related Topics and Future Directions

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

Forcing and Generic Models: We again assume all formulas are ∀-free. Let P 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 another forcing relation, for x ∈ X,

def. x P ϕ(a) ⇐==⇒ x ∈ kϕ(a)k .

Caicedo [Cai95] called “U ” the local semantics and “x ” the punctual semantics. On the other hand, each stalk Px is an L-structure, and we can also consider 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)| = kR(a)k. However, in general, |ϕ(a)| 6= kϕ(a)k. Some authors considered the relationship between them ( [Man77, §1] and [Lou79, Theorem 4.3, Lemma 5.1]). Kaiser [Kai77] addressed the problem when the relations Px |= ϕ(ax) and x P ϕ(a) coincide for any formula. He called such Px a generic stalk. If the ﬁlter nx is Θ(P )-generic, then Px is a generic stalk by our Ło´s-typetheorem. Kaiser used generic stalks to obtain omitting types and consistency results similar to those in [Kei73] (cf. [Cai95, §6], [BM04]). From a topos-theoretic perspective, Blass & Scedrov [BS83] constructed the classifying toposes of existentially closed models and ﬁnite-generic models. Their work was apparently inspired by Keisler’s viewpoint [Kei73] and might be related to ours.

Stalks, Global Sections, and Induced Geometric Morphisms: In addition to stalks of sheaves, the structure Γ(X,P ) of global sections is of our future interest (see below). The Feferman–Vaught theorem works for global sections just like Ło´s’stheorem does for stalks. Comer [Com74] gave a sheaf-theoretic interpretation of the original Feferman–Vaught theorem [FV59]. Feferman– Vaught type theorems and their applications to sentences preserved under taking global sections were pursued in [Vol76], [LL85], [Man77], [Tak80] and [BW79] (cf. [Vol79]). From a topos-theoretic viewpoint, taking stalks and global sections can be seen as part of geometric morphisms. Any morphism f : X → Y of locales (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) induced by the point f = x: 1 → X.

• The set P (X) is f∗P for the (essentially unique) geometric morphism Sh(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 Corollary 1.21. Therefore, we may investigate stalks and global sections in the more general framework of base change of Heyting-valued structures. This categorical approach has an advantage over the set-theoretic approach of [ACM19] to base change of Heyting-valued universes since the construction of geometric morphisms is much simpler and the logical behavior under base change along them is well-understood for various classes of morphisms of locales [Elephant, Chapter C3].

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

28 monograph [Kno12] includes a brief description of a history of sheaf representations of algebras (see also [Joh82, Chapter V]). Sheaf representations over Stone spaces, e.g., Pierce representation of commutative rings [Pie67], play a special role in model theory. Following Lipshitz & Saracino [LS73], Macin- tyre [Maci73] established a general method for obtaining model-companions of theories whose models have sheaf representations over Stone spaces with good stalks (cf. [Cars73]). He exploited Comer’s version of the Feferman– Vaught theorem to transfer model-theoretic properties of stalks to global sections. 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 models of well-behaved theories. For example, any (commutative) von Neumann regular ring R is represented by a sheaf of rings over a Stone space X(R) whose stalks are fields, and such a sheaf is a model of the theory of fields in the topos Sh(X(R)). The theory of von Neumann regular rings has the model- completion, whose models are represented by “algebraically closed fields” in sheaf toposes over Stone spaces. Thus, we may expect that developing model theory 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 the importance of the viewpoint we just mentioned. Our standpoint emphasizing Heyting-valued structures was greatly influenced by him too. Some other authors 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 model theory concerns theories internal to toposes, and internal theories in a sheaf topos admit sheaves of function symbols and relation symbols (cf. [Hen13]). It must be closer to doing model theory in a Heyting-valued universe (cf. [KK99]). The approach by Brunner & Miraglia [Bru16], admitting a presheaf of constant symbols in place of a set of constants, is regarded as a restricted form of topos-internal model theory. In contrast to the scarcity of research on topos- internal model theory, there is much more on universal algebra in toposes and sheaf models for constructive mathematics. Finally, we would like to mention a potential application of topos-internal model theory to algebraic geometry. At the end of [Lou79], Loullis suggests that algebraic geometry over von Neumann regular rings [SW75] could be obtained by doing algebraic geometry in some topos. The works of Bunge [Bun82] and her student MacCaull [MC88] reflect that idea, but no one followed them. We leave that direction as the ultimate goal of our research.

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