Elementary Topos Theory and Intuitionistic Logic

C.L. Mahany

August 28, 2012

Abstract

A topos is a particular kind of category whose definition has rich and strikingconsequences in various contexts. In this expository paper, the role that topoi playin intuitionistic logic is explored through Heyting algebras. In particular, I examinefirst the relationship between the axioms of intuitionistic propositional calculus andthe structure of a Heyting algebra, and then the relationship between the structureof a Heyting algebra and that of a topos. In the third section I derive propositionsnecessary for the main result. The main result is the proof of the existence of anatural, internal Heyting algebra in any topos.

Contents

1 Categorical necessities 1

2 Algebraic logic 7

3 Elementary topos theory 11

4 External and internal Heyting algebras 18

5 Conclusion and acknowledgements 22

References 23

1 Categorical necessities

We first introduce the requisite categorical concepts for discussing topos theory.

Definition 1.1. A category C consists of the following data:

• A collection of objects Ob(C);

• For any objects X,Y ∈ Ob(C), a collection C(X,Y ), called the morphisms from Xto Y . We write f : X → Y for a morphism f from X to Y.

1

• For any objects X,Y, Z ∈ Ob(C), a binary operation composition,− − : C(Y,Z)× C(X,Y )→ C(X,Z), such that for all morphismsf : W → X, g : X → Y and h : Y → Z, the following holds:

(h g) f = h (g f).

• For each object X ∈ Ob(C), an identity morphism 1X : X → X such that, for eachmorphism f : X → Y , we have f 1X = f = 1Y f.

The most familiar example of a category is Set, in which the objects are sets andthe morphisms are functions.

Definition 1.2. A covariant functor F : C → D between two categories C and D is anassignment to each object X ∈ Ob(C) of an object FX ∈ Ob(D), and to each morphismf : X → Y ∈ Mor(C) of a morphism Ff : FX → FY ∈ Mor(D) such that F (1X) = 1FXfor all X ∈ Ob(C) and F (g f) = F (g) F (f) for all f, g ∈ Mor(C).

Definition 1.3. For any category C, one can construct the dual category Cop by, forall objects X,Y ∈ Ob(C), interchanging the source with the target of each morphism inC(X,Y ); i.e., if f : X → Y in C then f : Y → X in Cop.

We say a covariant functor F : Cop → D is a contravariant functor F : C → D. Forcontravariant fuctors we have F (gf) = F (f) F (g) for all f : X → Y, g : Y → Z in C.

Definition 1.4. A natural transformation η : F → G between two functors F,G : C → Dis a collection of morphisms ηX : FX → GX for each X ∈ Ob(C) such that for anymorphism f : X → Y ∈ Morph(C), the following diagram commutes:

FX

Ff

ηX // GX

Gf

FYηY // GY

.

A natural transformation η is called a natural isomorphism if each of its components isan isomorphism, where an arrow f : X → Y is an isomorphism if there is some arrowg : Y → X for which fg = 1Y and gf = 1X . We call the morphism ηX : FX → GX thecomponent of η at X.

Lemma 1.5. (Yoneda) For any functor F : C → Set and for any object X of C, thereis a bijection

Nat(C(X,−), F ) ∼= FX

natural in both X and F, where C(X,−) denotes the functor from C into Set whichassigns to each object Y of C the set C(X,Y ), and Nat(C(X,−), F ) denotes the set ofnatural transformations from C(X,−) to F.

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Proof. The proof of this lemma rests upon both the diagram

C(X,X)φX //

f−

FX

Ff−

C(X,Y )φY // FY

and the observation that the identity morphism 1X : X → X must live in C(X,X).Consider a natural transformation φ : C(X,−) → F. For any f : X → Y, we have

that the above diagram is commutative. “Chasing” the obligatory identity morphismaround the diagram, we see that Ff(φX(1X)) = φY (f). We thus obtain the elementφX(1X) ∈ FX.

Conversely, observe that each element z ∈ FX determines a unique natural transfor-mation γz as follows. For each morphism f : X → Y, put γzY (f) = Ff(z). For instance,for the identity morphism 1X : X → X, we have γzX(1X) = F (1X)(z) = 1FX(z) = z. Itfollows from our choice of γz that γz is indeed a natural transformation. The twofoldnaturality of the bijections is mechanically verified as well.

The contravariant version of the Yoneda Lemma is proved analogously and statesthat Nat(C(−, X), F ) ∼= FX. The following will be useful later:

Proposition 1.6. The Yoneda Lemma respects (component-wise) composition of naturaltransformations, in the (contravariant) sense that the diagram

Nat(C(−, B), C(−, C))×Nat(C(−, A), C(−, B)) //

Nat(C(−, A), C(−, C))

C(B,C)× C(A,B)

// C(A,C)

commutes.

Proof. Suppose we have natural transformations α : C(−, B)→ C(−, C) andβ : C(−, A) → C(−, B). By the Yoneda Lemma, we have that α and β correspond toelements α = αB(1B) ∈ C(B,C) and β = βA(1A) ∈ C(A,B), and, composing thesemorphisms, we obtain αβ ∈ C(A,C). This corresponds to the bottom-left path of thediagram.

Alternatively, by the Yoneda Lemma, the natural transformation αβ : C(−, A) →C(−, C) obtained by composing at each component corresponds to an element αβ =(αβ)A(1A) = αA(βA(1A)). As βA(1A) is a morphism A→ B, we have by naturality of αthat the diagram

C(B,B)αB //

−βA(1A)

C(B,C)

−βA(1A)

C(A,B)αA // C(A,C)

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commutes, and so, chasing the identity 1B, we have that

αβ = αA(βA(1A))

= αA(1B βA(1A))

= αB(1B) βA(1A)

= αβ,

as required.

Definition 1.7. A contravariant functor F : C → Set is representable if it is naturallyisomorphic to C(−, A) for some object A of C. A representation of F is a pair (F, φ),where φ : C(−, A)→ F is a natural isomorphism.

Definition 1.8. A diagram of shape S in a category C is a functor D : S → C. A coneof the diagram D is an object N in C, equipped with morphisms Ni : N → Di for eachobject i in S, such that for any g : Di→ Dj, we have that gNi = Nj ; i.e., the followingdiagram commutes:

NNi //

Nj

Di

gDj

.

We say that the object N is the vertex of the cone.

Definition 1.9. The limit (unique up to unique isomorphism) of a diagram D of shapeS in a category C is a cone with vertex L ∈ Ob(C) such that for any other cone withsome vertex M ∈ Ob(C) there exists a unique morphism h : L→M such that Mi = hLi,for all objects i in the shape category S. Diagrammatically, the following must commute:

Mh //

Mi

L

Li~~Di

.

Intuitively, we can “factor” the cone with vertex M through the cone with vertexL. This distinguishing property for the limit — that any other cone can be factoredthrough it — is categorically referred to as a universal property, in the sense that thelimiting cone is universal among such cones.

Definition 1.10. The equalizer of two morphisms f, g : X → Y in a category C is thelimit of the diagram

X))55 Y .

For any two objects X,Y in a category C, the product of X and Y is the limit of thediagram which just has X and Y and their identity morphisms, and is denoted X × Y.

4

For any morphisms f : X → Z and g : Y → Z in some category C, the pullback of falong g or, equivalently, of g along f , is the limit of the diagram

Y

g

X

f // Z

.

Definition 1.11. A colimit in C is a limit in Cop, a coproduct is a product in Cop, acoequalizer is an equalizer in Cop, and a pushout is a pullback in Cop.

The following exercise will be of use in section 3.

Exercise 1.12. Any finite limit in a category C can be constructed from finite productsand equalizers in C.

Definition 1.13. An adjunction between two categories C and D is a pair of functors,F : C → D and G : D → C, together with, for each pair of objects X ∈ C, Y ∈ D, abijection

φX,Y : C(X,GY ) ∼= D(FX, Y ),

which is natural in X and Y. This twofold naturality requires both that for any morphismf : X → X ′ in C the diagram

C(X ′, GY )φX′,Y //

−f

D(FX ′, Y )

−Ff

C(X,GY )φX,Y // D(FX, Y )

commutes and that the analogous diagram for each map g : Y → Y ′ commutes. We sayin this situation that F is left adjoint to G, and G is right adjoint to F, and we denotethis relation by F a G.

Example 1.14. A class of examples of adjunctions concerns the relation between “free”and “forgetful” functors. For instance, the functor F : Set → Set∗, which adjoins adisjoint basepoint ∗A to A, is left adjoint to the “forgetful” functor U : Set∗ → Set,which forgets about the basepoint ∗A for each set A. Moreover, the free group functorF : Set→ Grp is left adjoint to the forgetful functor U : Grp→ Set, which forgets thegroup structure of groups.

Alternatively,

Definition 1.15. An adjunction between two categories C and D consists of a pair offunctors, F : C → D and G : D → C, and two natural transformations, η : 1C → GF andε : FG → 1D, such that εF Fη = 1F and Gε ηG = 1G; i..e, for any objects X andY in the categories C and D we have εFX F (ηX) = 1FX and G(εY ) ηGY = 1GY ,respectively. We say that η and ε are the unit and counit of the adjunction.

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Proposition 1.16. The two given definitions of adjunction between two categories Cand D are equivalent.

Proof. By φX,Y we mean the bijection between the hom-sets C(X,GY ) and D(FX, Y ).Given such bijections, we construct the unit η : 1C → GF by putting ηX = φX,FX(1FX)for all objects X of C, and ε : FG → 1D by putting εY = φ−1

GY,Y (1GY ) for all objects Yof D. The reader may readily verify that the naturality of η and ε readily follows fromthe naturality of the bijections. As ηX : X → GFX and εY : FGY → Y, the naturalityof the bijection shows that

εFX F (ηX) = φ−1X,FX(ηX) = φ−1

X,FX(φX,FX(1FX)) = 1FX ,

andG(εY ) ηGY = φGY,Y (εY ) = φGY,Y (φ−1

GY,Y (1GY ) = 1GY ,

as required.Conversely, suppose we have natural transformations η : 1C → GF and ε : FG→ 1D.

For any arrow f : X → GY , let φX,Y (f) = εY Ff, and for any g : FX → Y letφ−1X,Y (g) = Gg ηX . To verify that this indeed produces a bijection, observe that forf : X → GY and g : FX → Y we have from the naturality of ε and η that

φ−1X,Y (φX,Y (f)) = φ−1

X,Y (εY Ff) = G(εY Ff) ηX

= G(εY ) GFf ηX = G(εY ) ηGY f = 1GY f = f

andφX,Y (φ−1

X,Y (g)) = φX,Y (Gg ηX) = εY F (Gg ηX)

= εY FGg F (ηX) = g εFX F (ηX) = g 1FX = g.

The naturality of these bijections follows from the naturality of η and ε.

A noteworthy property of adjoints is that every left adjoint preserves limits and everyright adjoints preserves colimits.

Definition 1.17. A monomorphism or monic m : Y → Z in C is a morphism from Yto Z such that for any two morphisms f, g : X → Y, we have that mf = mg impliesf = g. We say that two monomorphisms m : X → Z and n : Y → Z are equivalent ifthere exists an isomorphism h : X → Y such that the diagram

X

h

m // Z

Y

n

>>

commutes. A subobject of an object Z in C is an equivalence class of monomorphismsinto Z.

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Definition 1.18. For any two objects X,Y in a category C, the exponential of X andY consists of an object Y X and an arrow e : Y X×X → Y such that for any other objectZ and arrow g : Z ×X → Y , there is a unique f ′ : Z → Y X such that e (f ′ × 1X) = f ;i.e., making the diagram

Y X ×X e // Y

Z ×X

f ′×1X

OO

f

;;

commute. Thus, if we have an exponential Y X for all objects X,Y in the category C,then we have an isomorphism C(Z×X,Y ) ∼= C(Z, Y X), so that the functor −×X is leftadjoint to −X .

Definition 1.19. An initial object of a category C is a limit of the empty diagram, whilea terminal object is a colimit of the empty diagram. A category C is cartesian closed ifit has a terminal object and for all objects X,Y ∈ Ob(C) both a product X × Y and anexponential Y X .

Definition 1.20. For any object X in a category C, the slice category C/X is thecategory with morphisms into X from C as its objects; for any two objects a : A → Xand b : B → X of C/X, a morphism f : a → b in C/X is an arrow f : A → B such thatbf = a.

2 Algebraic logic

Definition 2.1. A lattice L is a partially ordered set which has for all x, y ∈ L botha supremum (or join) x ∨ y ∈ L and an infinum (or meet) x ∧ y ∈ L. A lattice L isdistributive if for any x, y, z ∈ L, we have that x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z). In latticeswith elements 0 and 1 such that 0 ≤ x ≤ 1, for all x ∈ L, a complement for an elementx is an element a ∈ L such that x ∧ a = 0 and x ∨ a = 1. A Boolean algebra B is adistributive lattice with 0 and 1 in which every element has a complement.

For any poset P, we can construct a category CP as follows. Let the objects of CPconsist precisely of the elements of P, and let there be a single morphism x→ y wheneverx ≤ y in P. If the poset is a lattice, then the category will have binary products andcoproducts, and if it is Boolean, then the category will have an initial and a terminalobject, namely 0 and 1.

The essential ingredients of propositional logic are propositional variables. Intuitively,they are variables which are either true or false. For any set of propositional variables wehave the binary operations “or”, “and”, and “implies”, denoted ∨,∧, and →; the unaryoperation, negation, denoted ¬; and the nullary operations > and ⊥ which correspondto true and false.

Definition 2.2. For a set P of propositional variables, we recursively define formulae,so that

7

• p is a formula, for all p ∈ P;

• Given any formula F, the negation ¬F is a formula;

• For any formulae F and G, each of F ∨G,F ∧G and F → G is a formula.

Definition 2.3. For a set of propositional variables P and a Boolean algebra B, avaluation is a function V : P→ B such that

V (>) = 1, V (⊥) = 0, V (P ∧Q) = V (P ) ∩ V (Q), V (¬F ) = ¬V (F ),

V (P ∨Q) = V (P ) ∪ V (Q) and V (P → Q) = V (P )⇒ V (Q),

for all formulae P and Q, where ∩,∪, and⇒ are the operations and ¬ is complementationin B. A formula F is valid if V (F ) = 1, for all valuations V into 0, 1. A formula F isB-valid if V (F ) = 1 for every valuation into B.

The completeness property of classical propositional logic states that a formula F isprovably true iff F is valid — i.e., F is always “true.” However, noting that the proofof this fact only relies on algebraic properties satisfied by all Boolean algebras, and notjust 0, 1, we can state instead:

Theorem 2.4. Any formula F is provable in classical propositional logic if and only ifF is B-valid for every Boolean algebra B.

This is the precise sense in which Boolean algebras serve as models for classicalpropositional calculus. A hallmark feature of intuitionistic logic is its exclusion of thenotorious “law of excluded middles,” that > = p ∨ ¬p, for all p ∈ P. A striking con-sequence of this limitation is the invalidity of proofs by contradiction. Analogously,Heyting algebras — in which neither p nor ¬p are necessarily true — serve as modelsfor intuitionistic propositional calculus (henceforth referred to as IPC).

Definition 2.5. A Heyting algebra H is a poset with all finite products and coproductswhich, considered as a category, is cartesian closed. Analogously to Definition 2.3 above,a valuation into H is a structure-preserving function V : P → H and a formula F is H-valid if V (F ) = 1 for every valuation into H. For a finite set of formulas Γ, we let

∧Γ

denote the formula obtained by taking the join over Γ. A formula F is an H-consequenceof Γ if V (

∧Γ) ≤ V (F ) for every valuation into H.

Definition 2.6. For a set Γ of formulae, we write Γ ` F in IPC if there exists a deductionending with F that only uses IPC axioms, elements of Γ, and the inference rule, modusponens:

F, (F → G)

G,

to be read G is deduced from F and F → G.In particular, the IPC axioms are

F → (G→ F ),

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(F → (G→ H))→ ((F → G)→ (F → H)),

F ∧G→ F, F ∧G→ G,F → (G→ (F ∧G)),

F → F ∨G,G→ F ∨G,

(F → H)→ ((G→ H)→ (F ∨G→ H)),

and⊥ → F,

for all formulae F,G, and H.

Definition 2.7. A formula F is provable in IPC if there exists a derivation > ` F , wherea derivation from > is a sequence of applications of modus ponens which concludes withF and uses only axioms of IPC.

We can model logic with a lattice by taking the elements of the lattice to be allformulae, and by putting F ≤ G if and only if > ` F → G, for all formulae F and G.A Heyting algebra H can thus be equivalently defined as a lattice with 0 and 1 whichhas for all elements a, b ∈ H an exponential ba, denoted a ⇒ b, which is characterized— under the exponential adjunction — by c ≤ (a⇒ b) if and only if c ∧ a ≤ b. That is,the element a⇒ b is the least upper bound among all such elements c. Note that everyBoolean algebra is also a Heyting algebra, as c ≤ (¬a ∨ b) if and only if c ∧ a ≤ b.

If F is an element of Γ then Γ ` F, trivially. If not, then, by the definition of aformula, we must have that F is the product of some sequence of operations defined onpropositional variables. The following proof is motivated by the observation that anyderivation Γ ` F must end with an assertion of the structure of F :

F = G ∨ J, Γ ∨ G ` F, Γ ∨ J ` F, Γ ` G ∨ J ;

F = G ∧ J, Γ ` G, Γ ` J ;

or F = (G→ J), Γ ∨ G ` J.

Lemma 2.8. If Γ ` F in IPC then F is an H-consequence of Γ for every Heytingalgebra H.

Proof. We show this by induction by considering all three possibilities for the last stepin the deduction Γ ` F , as described above.

If F = G ∧ J, for some formulae G and J, then we must have that Γ1 ` G andΓ2 ` J for some subsets Γ1 and Γ2 of Γ. Hence, our induction hypothesis states thatV (

∧Γ1) ≤ V (G) and V (

∧Γ2) ≤ J. From the definition of V, we consequently have that

V (∧

Γ) = V (∧

Γ1 ∧∧

Γ2) = V (∧

Γ1) ∧ V (∧

Γ2)

≤ V (G) ∧ V (J) = V (G ∧ J) = V (F )

so that F is indeed an H-consequence of Γ.

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If F = G ∨ J for some formulae G and J, then there are derivations Γ ` G ∨ J,Γ ∨ G ` F and Γ ∨ J ` F. Thus, from our induction hypothesis, we have that

V (∧

Γ) ≤ V (G ∨ J) = V (G) ∨ V (J),

V (∧

Γ) ∨ V (G) ≤ V (F ),

and V (∧

Γ) ∨ V (J) ≤ V (F ).

Consequently, we obtain

V (∧

Γ) ≤ V (∧

Γ) ∧ (V (G) ∨ V (J))

= (V (Γ) ∧ V (G)) ∨ (V (Γ) ∧ V (J)) ≤ V (F ) ∨ V (F ) = V (F ∨ F ) = V (F ).

Finally, if F = (G→ J) then Γ∨ G ` J, so we have from the induction hypothesisthat

V (∧

Γ ∪ G) = V (∧

Γ) ∧ V (G) ≤ V (J).

By the definition of ⇒ in a Heyting algebra, we therefore have that V (∧

Γ) ≤V (G)⇒ V (J) = V (G→ J) = V (F ), as desired.

Theorem 2.9. (Completeness for IPC) Any formula F is provable in IPC if and onlyif F is H-valid in every Heyting algebra H.

Proof. Lemma 2.8 nearly completes the forward direction; if F is provable in IPC, then> ` F, and so, by Lemma 2.8, we have V (>) = 1 ≤ V (F ), and, as 1 is the top elementof any Heyting algebra H, we also have V (F ) ≤ 1. Therefore, we have that V (F ) = 1,so that F is H-valid in every Heyting algebra H.

For the converse, we must construct a certain Heyting algebra and apply a particularvaluation. Denote by F the set of all formulae. We can then define an equivalencerelation ∼ in F by putting G ∼ J when and only when ` G ⇔ J ; that is, when wecan deduce both G ⇒ J and J ⇒ G from the axioms of IPC. Let H = F/∼. For anyelements [F ], [G] ∈ H, we let [F ] ≤ [G] when and only when ` F ⇒ G. For the top andbottom elements, we let 1 = [>] and 0 = [⊥]. For operations in the Heyting algebra,we let [F ] ∧H [G] = [F ∧ G], [F ] ∨H [G] = [F ∨ G], and [F ] ⇒H [G] = [F ⇒ G]. Thereader may readily verify that these operations are well-defined, and that they indeedsatisfy the axioms for a Heyting algebra. We now let our valuation V : P→ H be givenby V (J) = [J ], for all propositional variables J ∈ P.

Thus, for a formula F valid in H, we have V (F ) = [F ] = 1 = V (>) and so, as> ∈ 1, we have that F ∼ >. Therefore, by our definition of ∼, we have ` F ⇔ >, andso > ` F.

We are thus licensed to assert that giving a valuation into a Heyting algebra forwhich V (F ) = 0 is enough to ensure that F is not provable in IPC.

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3 Elementary topos theory

Definition 3.1. A topos E is a category with

• All finite limits;

• An object Ω and a monic true : A → Ω such that for any monic m : S → X thereis a unique arrow φ : X → Ω in E rendering the square

S //

m

A

true

Xφ // Ω

a pullback. We call φ the classifiying map of m and Ω the subobject classifier of E .

• For every object X, an object PX and an arrow ∈X : X × PX → Ω such thatfor any arrow f : X × Y → Ω there is a unique arrow f ′ : Y → PX for which thediagram

X × Y f //

1×f ′

Ω

X × PX ∈X // Ω

commutes.

It is an exercise to show that every topos has exponentials, which are constructedfrom finite limits and power objects. It is a bit more involved to show that every toposhas all finite colimits, which is a consequence of the fact that the power-set functor ismonadic. Both of these results are illustrated in section IV of [2].

Notice that the second and third requirements of the above definition give us bi-jections SubE(Y ) ∼= E(Y,Ω) and E(X × Y,Ω) ∼= E(Y, PX), respectively. The fact thatSubE(−) and E(X ×−,Ω) are representable functors will allow us to thoroughly utilizeYoneda’s Lemma and Proposition 1.6.

Proposition 3.2. In a topos, the source A of the monic true is the terminal object 1.

Proof. We begin by taking the pullback of true along true, as on the left:

Pp //

p

A

true

Atrue // Ω

; Xq // A

true

Xφ1x // Ω

.

For any object X, note that 1X is a monic, and so there exists some q : X → A suchthat the diagram above on the right is a pullback square. To show the uniqueness of

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this q and that A is therefore the terminal object, suppose that there are two morphismsq, q : X → A. Note that both diagrams

Xq // A

true

Xtrue q // Ω

Xq′ // A

true

Xtrue q′ // Ω

are trivially pullbacks, and so, by the uniqueness of the classifying map for the subobject1X : X → X, we have that true q = true q′ and so q = q′, as true is monic.

Definition 3.3. The image of the arrow f : X → Y is a monomorphism m : M → Ytogether with an arrow e : X → M such that f = me and if f factors through anyother monic k, then m does as well; that is, the image is universal among such factoringmonomorphisms.

Notice the similarity of universal categorical properties — such as the image — and“universal” poset properties, such as the least upper bound. Indeed, we will need to usethe image in defining the supremum of our lattice.

Proposition 3.4. In a topos, every arrow f has an image m.

Proof. For any arrow f : A → B, consider the pair of arrows x, y : B → P in the push-forward of f along f

Af //

f

B

y

B

x // P

,

and let m : M → B be the equalizer of x and y. Suppose that mh = mh′ for some objectH and arrows h, h′ : H → B. Then the diagram

H

h

mh

M

m // Bx ))

y55 P

H

h′

OO

mh′

>>

commutes and by the universality of the equalizer m we must have that h = h′, so that mis monic. As f : A→ B we have also by this universality a unique morphism e : A→Msuch that f = me.

Note that in a topos every monic m : S → B is the equalizer of some pair of arrows —in particular, of true !B and φm, where !B is the unique map !B : B → 1 and φm : B → Ωis the classifying map for m. The universality of this equalizer is a consequence of theuniversality of the pullback square of true!B along φm. Hence, for any monic h : H → B

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through which f factors as, say, f = hq for some q : A → H, we have that h is anequalizer of some pair of arrows s, t into some object P ′. Thus, sf = shq = thq = tf,and so by the universality of the pushforward P we have a unique arrow j : P → P ′ suchthat jx = s and jy = t. Consequently we have that sm = jxm = jym = tm and so bythe universality of the equalizer h there is a unique l : M → H such that m = hl; thatis, m factors through h, as desired.

Let X be an object of the topos E . Given two subobjects m : M → X and n : N → Xof X, we say that m ≤ n (more precisely [m] ≤ [n]) when there is some arrow g : M → Nof E such that

Mg //

m

N

n~~X

commutes. Observe that ≤ gives a poset SubE(X).

Lemma 3.5. For every object X in a topos E , the poset SubE(X) is a lattice.

Proof. We must, as per Definition 2.1, show that SubE(X) has all finite meets and joins.Define the meet of any two subobjects m : M → X and n : N → X to be the uniquearrow m ∩ n : M ∩N → X defined by the pullback

M ∩N //

N

M // X

,

which indeed is a monic into and therefore subobject of X, as pullbacks preserve monicsand the composition of monics is a monic. The universal property of the pullback in Eis precisely the requirement in the poset that m ∩ n : M ∩N → X is the greatest lowerbound of m and n.

For the join of m and n, consider the diagram

M +N

##

Nq2oo

n

E

m∪n M

q1

OO

m// X

,

where M + N denotes the coproduct of M and N and m ∪ n : E → X is the image ofthe unique arrow m+n from M +N to X for which (m+n) q1 = m and (m+n) q2 = n.As the image m ∪ n is monic, it is a subobject of X. Its universality in E is equivalent,in the the poset SubE(X), to the condition of being a least upper bound.

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Definition 3.6. For each f : X × Y → Ω, the P-transpose of f is the unique mapf ′ : Y → PX for which ∈X (1×f ′) = f. The diagonal map is the unique map ∆X : X →X ×X for which π1∆X = π2∆X = 1X . It is trivially a monic and thus, as a subobjectof X ×X, classified by a unique map δX : X ×X → Ω. The P -transpose of δX is a map·X : X → PX which we call the singleton map of X.

Definition 3.7. Given a morphism f : X → Y in a topos E , we define the arrowPf : PY → PX to be the unique map such that the diagram

X × PY f×1 //

1×Pf

Y × PY∈Y

X × PX ∈X // Ω

commutes.

Proposition 3.8. The above assignment f // Pf defines a contravariant functorP (−) : E → E .

Proof. We need to show for any arrows and objects f : X → Y, g : Y → Z in E thatP (gf) = PfPg, which is equivalent to the commutativity of the diagram

X × PZ f×1 //

1×Pg

Y × PZ g×1 // Z × PZ∈Z

X × PY 1×Pf // X × PX ∈X // Ω

.

From the definitions of Pf and Pg, we have that

∈X (1× Pf)(1× Pg) =∈Y (f × 1)(1× Pg) =∈Y (f × Pg)

=∈Y (1× Pg)(f × 1) =∈Z (g × 1)(f × 1),

and so P (gf) = PfPg.

We are working towards showing that SubE(X) is a Heyting algebra, for any objectX. However, it is far easier to show that SubE(1) is a Heyting algebra, where 1 is aterminal object. Given the isomorphism SubE/X(1X) ∼= SubE(X), we can far morereadily conclude that SubE(X) is a Heyting algebra if we can show that E/X is a topos.Interesting in its own right, we therefore need the theorem

Theorem 3.9. For any object X in a topos E , the slice category E/X is also a topos.

Proof. Given two objects a : A→ X and b : B → X of E/X, consider the pullback

A×X Bp2 //

p1

B

b

Aa // X

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in the topos E . Note that the composite 〈a, b〉 = ap1 = bp2 : A ×X B → X is (up tounique isomorphism) the product of a and b in E/X, as any object q in E/X with mapsq1 : q → a and q2 : q → b in E/X gives rise to a unique map j : q → 〈a, b〉 from theuniversality of the pullback in E . Similarly, note that for any pair of arrows h, h′ : a→ bin E/X, the equalizer e is simply the composite he = h′e, where e : E → A is the equalizerof h and h′ in E . From Proposition 1.12 we thus have that the slice category E/X hasall finite limits.

To see that E/X has a subobject classifier, note first that 1X is the terminal objectin E/X. As the identity 1X : X → X is monic, it is classified in E by a unique map v forwhich v = true !X . Let trueX = true !X×1X : X → Ω×X, and let Ω0 = π2 : Ω×X → X,where π2 is the second projection. Any monic γ : a → b in E/X, where a : A → X andb : B → X, is a monic γ : A→ B in E and therefore a subobject of B. It is thus classifiedby a unique map φγ which renders the diagram

A!A //

γ

1

true

Bφγ // Ω

a pullback. Consequently, for any monic γ : a→ b, the map γ = φγ × b : B → Ω×X isa morphism γ : b→ Ω0 in E/X such that the diagram

a

γ

// 1X

trueX

bγ // Ω0

is a pullback, as any map η : c → b for some c : C → X in E/X yields a unique mapη′ : C → A by the universality of the pullback in E . We have therefore shown that Ω0,together with the monic trueX = true !X × 1X , is a subobject classifier for E/X.

Showing that E/X has power objects amounts to providing for each object a of E/Xan object PXa such that E/X(〈a, b〉,Ω0) ∼= E/X(b, PXa) for each object b : B → X ofE/X. Notice first that A ×X B is a subobject of the product A × B in E , because thepullback of the monic ∆X along a×b : A×B → X×X gives a monic m : A×XB → A×B.

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This subobject is therefore classified by a unique map A×B → Ω. The diagram

A×X B //

m

X

∆X

// 1

true

A×B a×b // X ×X δX // Ω

A×B 1×b // A×X a×1 // X ×X1×·X// X × PX ∈X // Ω

A×X1×·X // A× PX

a×1

OO

1×Pa // A× PA

∈A

OO

A×B 1×w // A× PA

,

whose commutativity follows from the definitions of the morphisms involved, shows thatthe subobject m is classified by ∈A (1× w) =∈A (1× Pa ·X b) : A×B → Ω.

Hence, there is a bijection E/X(〈a, b〉,Ω0) ∼= E(A×X B,Ω), under which morphismsz : A×X B → Ω0 get sent to π1z, and morphisms y : A×X B → Ω get sent to y× 〈a, b〉.E(A×XB,Ω) is isomorphic to SubE(A×XB), which is in turn isomorphic to s : S →

A × B | s ∈ SubE(A × B) and s ≤ m, where s ≤ m means s = mn for some n. Thisbijection can be seen by noting that each such n is a subobject of A ×X B and thatany such subobject uniquely yields a subobject s = mn of A×B, as the composition ofmonics is a monic.

Observe that s ≤ m iff S ∩ A×X B = S, where ∩ denotes the intersection operatorof the lattice E(A × B,Ω). If S ∩ A ×X B = S then we must have that mn = s1S = sfor some n : S → A×X B, and if mn = s for some n then 1S must be the pullback of malong s. Consequently, we have a further bijection to the set

h : A×B → Ω |h ∧ ∈A(1× w) = j,

as each subobject of A×B is uniquely classified by a map h : A×B → Ω, and h∧∈A(1×w) = h if and only if S ∩A×X B = S.

Taking the P -transpose of each h provides us with yet another bijection, to the setk | k : B → PA and k∧w = k. From the diagram above, we have that w = Pa·X b,and so

k = k ∧ w = ∧(k × w) = ∧(k × (Pa ·X b))

= ∧(1× (Pa ·X))(k × b) = t(k × b)

where t = ∧(1× Pa·X). This gives a bijection to the set of those arrows k : B → PAfor which π1(k × b) = k = t(k × b). Let PXa be the equalizer of π1 and t, as in the

16

diagram

Bk×b //

PA×Xπ1

,,

t

22

1×·X

PA

PXa

e

99

PA× PX1×Pa

// PA× PA

∧

OO .

Our succession of bijections shows that E/X(〈a, b〉,Ω0) ∼= k : B → PA | p(k×b) = t(k×b), and the universality of the equalizer PXa gives a further bijection into E/X(b, PXa).Thus, the category E/X also has power objects, and so E/X is a topos, for any objectX.

To say that the Heyting algebra structure is natural in X intuitively means thatgiven some f : X → Y in E , it does not matter whether we first perform operations onsubobjects of Y and then convert them to subobjects of X, or vice versa. Essentially itamounts to the property that our Heyting algebras SubE(X) are not biased by our choiceof X, and that the Heyting algebra structures respect, to some extent, the structure of E .As any element of SubE(X) is an object of E/X, it turns out that proving the followingfor any pullback functor f∗ : E/Y → E/X will suffice.

Theorem 3.10. For any morphism f : X → Y in the topos E , the functor f∗ : E/Y →E/X, defined by pullback along f , has both a left and a right adjoint and preservesexponentials in the sense that f∗(mn) ∼= f∗mf∗n, for all objects m,n of E/Y.

Proof. We first prove the result for Y = 1, that is, pullback along morphisms f : X → 1,so that f∗ : E/1→ E/X. Notice that the functor f∗ here is simply −×X, as the productB×X is the pullback of !B along f, where !B is the unique map B → 1. By the definitionof the exponential, we thus have a right adjoint −X to −×X = f∗.

Define the functor f : E/X → E/1 to send an object l : L → X to the unique map!L : L → 1. For any objects l : L → X and !M : M → 1 of E/X and E/1, respectively,we have a bijection E/1( !L, !M ) ∼= E/X( l , f∗ !M ). This is because arrows h : l → f∗!Min E/X are morphisms h : L → M × X in E for which π2h = l, and so they uniquelydetermine a morphism π1h : L→M in E and, as !Mπ1h : L→ 1, we have !Mπ1h =!L, sothat π1h : !L →!M in E/1. Conversely, each arrow j : !L →!M is a morphism j : L → Min E , and it uniquely determines a map j × l : L→M ×X, for which π2(j × l) = l andso j × l : l→ f∗m. That is, f a f∗.

To see that f∗ preserves exponentials, note first that f∗(mn) ∼= f∗mf∗n if and onlyif the diagram

E/1(−)n //

f∗

E/1

f∗

E/X

(−)f∗n// E/X

commutes, for all objects m,n in E/1. However, recall from the proof of Theorem 3.9that the pullback in the topos of objects from the slice topos is precisely the product in

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the slice topos, and, by the definition of the exponential, we therefore have left adjoints−×n a (−)n and −×f∗n a (−)f∗n. From the above, we also have the left adjoint f a f∗.Notice that we have an isomorphism between the compositions of these left adjoints, asindicated by the commutative diagram

P //

f∗N

f∗n

// N

n

K

k // Xf // 1

,

in which both squares are pullbacks, so that the entire rectangle is a pullback. Inparticular, the diagram shows that fk×n = fk×n ∼= f(k× f∗n), for any objects k andn of E/X and E/1, respectively.

As in the square at the bottom of the previous page, we have that the composite(−)f

∗n f∗ is right adjoint to f (− × f∗n), and so, by the isomorphism just shown,it is also right adjoint to (−× n) f , which in turn is left adjoint to f∗ (−)n. By theuniqueness of left adjoints, we therefore have that f∗ (−)n ∼= (−)f

∗n f∗, so that f∗

preserves exponentials.For any function f : X → Y — not necessarily with Y = 1 — notice that f is an

object in E/Y , and that an object in the slice topos (E/Y )/(f) is preicsely an arroww : M → X, for which fw = m. Every object w : M → X of E/X uniquely determinesthe arrow fw of (E/Y )/(f). We thus have a canonical isomorphism (E/Y )/(f) ∼= E/Xand a pullback functor f : (E/Y )/(1Y )→ (E/Y )/(f) which is naturally isomorphic to f∗.By the above, this functor f has both adjoints and preserves exponentials, and thereforeso does the isomorphic f∗.

4 External and internal Heyting algebras

Definition 4.1. A homomorphism of Heyting algebras H and H ′ is a morphism m : H →H ′ which preserves ∧,∨, and ⇒, as well as both top and bottom elements. This meansthat both diagrams

H ×H ∧H //

m×m

H

m

H ′ ×H ′∧H′ // H ′

, 1>H // H

m

1>H′ // H ′

commute, as do the analogous diagrams with ∨ and ⇒ or ⊥ in place of ∧ or >.

Our tedious work above now allows us to draw some surprising connections betweentopoi and IPC.

Theorem 4.2. For any object X in a topos E , the set SubE(X) is a Heyting algebra.This structure is natural in X, in the sense that for any arrow f : X → Y in the topos E,the functor f−1 : SubE(Y )→ SubE(X), defined by pullback along f , is a Heyting algebrahomomorphism.

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Proof. We have already shown in Lemma 3.5 that SubE(X) is a lattice. We will firstconcern ourselves with the lattice of subobjects of the terminal object, SubE(1). Notethat for any subobjects u : U → 1, v : V → 1, the objects U and V are in the toposE and consequently there is an exponential UV , and a unique arrow into the terminalobject, uv : UV → 1. Let f and g be morphisms from some object S to UV . Underthe (exponential) bijection E(S,UV ) ∼= E(S × V,U), we have for f and g correspondingmorphisms f, g : S × V → U. If uvf = uvg, then uf = ug, as in

S × Vf**

g

44 Uu // 1 .

However, u is monic, so we have that f = g, and so f = g under the bijection. Thus, uv

is monic and therefore is in SubE(1), which is consequently a Heyting algebra.If 1 is the terminal object of the topos E , and X is some object of the topos, then we

have a bijection SubE(X) ∼= SubE/X(1X). By Theorem 3.9, E/X is a topos, and so bythe above SubE/X(1X) is a Heyting algebra. Thus, under the isomorphism SubE(X) ∼=SubE/X(1X), we have that SubE(X) is a Heyting algebra, for any object X of the toposE .

As pullbacks preserve monics, we have that the pullback of any subobject m : M →Y of Y along f : X → Y produces a subobject of X. Let iY : SubE(Y ) → E/Y andiX : SubE(X)→ E/X be the canonical inclusions, i.e. iY (m : M → Y ) = m ∈ Ob(E/Y ).By the commutative diagram

SubE(Y )f−1//

iY

SubE(X)

iX

E/Y f∗ // E/X

and Theorem 3.10, we have that f−1 preserves limits, and therefore pullbacks, andtherefore the operation ∪. Notice from our proof of Lemma 3.5 that the operation ∩ isthe image of the coproduct, which is, from our proof of Proposition 3.4, an equalizer ofa pushout. As f∗ has a left and right adjoint, it preserves all finite limits and colimits,so we have that f−1 must preserve ∩. As we defined x ⇒ y to be the exponentialyx, and as f∗ preserves exponentials by Theorem 3.10, we have that f−1 too preservesexponentials. As the pullback of the bottom element !Y : 0 → Y , where 0 is the initialobject, is the bottom element !X : 0 → X of SubE(X), and as the pullback of the topelement 1Y : Y → Y is the top element 1X : X → X of SubE(X), we consequently havethat f−1 is a Heyting algebra homomorphism.

The result is significant because it demonstrates that every topos provides us with amodel of IPC; namely, SubE(1). Many logicians and mathematicians are concerned withnot just the law of excluded middles but various axioms of set theory, and it is by thistoken desireable to have an “internal” or Set-free analog of a Heyting algebra. The hope

19

is that, by working with internal objects, we might cleanse ourselves of any Set-specificlogical pathologies. By an external Heyting algebra, we mean an object in Set whichsatisfies Definition 2.5.

Definition 4.3. An internal lattice in a topos E is an object L equipped with twoarrows, meet ∧ : L× L→ L and join ∨ : L× L→ L, such that the diagrams expressingthe associative, commutative, and idempotent laws for both ∨ and ∧ as well as theabsorption law X ∧ (Y ∨ X) = X = (X ∧ Y ) ∨ X are commutative; see the belowdefinition for an example of such a diagrammatic expression. An internal lattice L hastop > and bottom ⊥ elements if there are arrows > : 1 → L and ⊥ : 1 → L from theterminal object of E such that ∧(1×⊥) = ∨(1×>) = 1L.

Definition 4.4. An internal Heyting algebra in a topos E is an internal lattice H withtop and bottom elements with an additional binary operation⇒ : H×H → H renderingcommutative the diagrams expressing the laws

X ⇒ X = 1, X ∧ (X ⇒ Y ) = X ∧ Y, Y ∧ (X ⇒ Y )

andX ⇒ (Y ∧ Z) = (X ⇒ Y ) ∧ (X ⇒ Z).

For example, the last equation is diagrammatically expressed as

H ×H ×H 1×∧ //

δ×1×1

H ×H

⇒

H ×H ×H ×H1×τ×1

H ×H ×H ×H⇒×⇒

H ×H ∧ // H

,

in which τ : H ×H → H ×H is the twist map, for which π1 = π2τ and π2 = π1τ ;δ : H → H × H is the diagonal map; and by ⇒(X,Y ) and ∧(X,Y ) we mean X ⇒ Yand X ∧ Y, respectively.

We also want internal Heyting algebras because each one gives rise to a whole familyof external Heyting algebras, through the bijection SubE(X × Y ) ∼= E(Y, PX). We wantnaturality internally so that the map Pf : PY → PX preserves the structure of PY, i.e.,so that ∧ (Pf × Pf) = Pf ∧.

Theorem 4.5. For any object X in a topos E , the power object PX is an internalHeyting algebra. For any morphism f : X → Y in E, the corresponding map Pf : PY →PA is a homomorphism of internal Heyting algebras. For each object Y in E, the setE(Y, PX) is an (external) Heyting algebra.

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Proof. From the previous theorem we have that Sub(X × B) is an external Heytingalgebra, for any objects X and B, and hence there is some meet operation ∩ : Sub(X ×B)×Sub(X×B)→ Sub(X×B). Theorem 4.2 shows that ∩ : Sub(X×B)×Sub(X×B)→Sub(X ×B) is natural in B, and so we have that the map ∧B in the diagram

Sub(X ×B)× Sub(X ×B)∩ //

Sub(X ×B)

E(B,PX)× E(B,PX)

∧B // E(B,PX)

is also natural in B. We therefore have a natural transformation ∧− from E(−, PX) ×E(−, PX) to E(−, PX). By the Yoneda Lemma, this natural transformation uniquelycorresponds to a morphism ∧ : PX × PX → PX; this ∧ is the meet operation for PX.We analogously obtain the operations ∨ and ⇒ : PX × PX → PX from the YonedaLemma and under the isomorphism SubE(X ×B) ∼= E(B,PX), as Theorem 4.2 ensuresthat ∨ and ⇒ are also natural in B.

Theorem 4.2 also shows that both >,⊥ : 1 → SubE(X × B) are natural in B, sowe have a natural transformation >− : ∗ → E(−, PX), where the functor ∗ : E → Setsends every object to the singleton set, ∗. As ∗ ∼= E(−, 1), the Yoneda Lemma uniquelydetermines a corresponding arrow>′ : 1→ PX. The bottom element is likewise obtained.

That these operations satisfy the requirements of an internal Heyting algebra is animmediate consequence of Theorem 4.2 and Proposition 1.6. To see this, consider thediagram

PX PX × PX∧oo

PX × PX

π1

OO

π1

δ×1// PX × PX × PX τ×1 // PX × PX × PX∧×1

1×∨

OO

PX PX × PX∨oo

and the isomorphisms

E(PX × PX,PX) ∼= Nat(E(−, PX × PX), E(−, PX))

∼= Nat(E(−, PX)×E(−, PX), E(−, PX)) ∼= Nat(E(X×−,Ω)×E(X×−,Ω), E(X×−,Ω))

∼= Nat(SubE(X ×−)× SubE(X ×−),SubE(X ×−)).

The operation ∧, for instance, thus corresponds to the natural transformation

∩ : SubE(X ×−)× SubE(X ×−)→ SubE(X ×−).

Proposition 1.6 tells us that the above diagram commutes for these morphisms on PX ifand only if it commutes for the corresponding natural transformations on SubE(X ×−).Commutativity of the corresponding natural transformations is precisely the commuta-tivity of the components, for every object. But for every object Y, we have by Theorem

21

4.2 that SubE(Y ×X) is a Heyting algebra, in which it is true that x ∧ (y ∨ x) = x =(x ∧ y) ∨ x. Hence, each necessary diagram commutes, including the above, and PX isan internal Heyting algebra.

To see the naturality of this internal structure, consider the diagrams

PY

Pf

PY × PYPf×Pf

∧oo

PX PX × PX∧oo

, SubE(Y ×−)

f×1−1

SubE(Y ×−)× SubE(Y ×−)

(f×1)−1×(f×1)−1

∩oo

SubE(X ×−) SubE(X ×−)× SubE(X ×−)∩oo

and the sequence of isomorphisms connecting them:

E(PY, PX) ∼= Nat(E(−, PY ), E(−, PX))

∼= Nat(E(Y ×−,Ω), E(X ×−,Ω)) ∼= Nat(SubE(Y ×−), SubE(X ×−)).

The left square commutes if and only if the right square does, which in turn commutesif and only if, for every object Z in the topos E , the diagram

SubE(Y × Z)

(f×1)−1

SubE(Y × Z)× SubE(Y × Z)

(f×1)−1×(f×1)−1

∩oo

SubE(X × Z) SubE(X × Z)× SubE(X × Z)∩oo

does. However, by Theorem 4.2, we have that (f × 1)−1 is a Heyting algebra homo-morphism, and so each diagram displayed commutes, as do the analogs for ∨,⇒,>, and⊥. Therefore, for any f : X → Y, we have that Pf : PY → PX is an internal Heytingalgebra homomorphism, so that the structure PX is natural in X. For any object Y ofE , we have that E(Y, PX) is an external Heyting algebra by the canonical isomorphismE(Y, PX) ∼= SubE(X × Y ).

5 Conclusion and acknowledgements

In effect, we have shown that there is an intimate relationship between topoi and Heytingalgebras, which we have seen are models for IPC. Anything that can be proven withoutthe law of excluded in middles — in IPC — must get sent to 1 under every valuationinto every Heyting algebra that we can obtain from a topos.

I can not possibly express how grateful I am for my mentor Michael Smith and all ofhis time that I wasted. He informed me early on that topos theory was rather difficult,but he was incredibly patient, without fail, throughout all of my struggles. Without hisgenuine concern for my studies and his remarkable ability to explain category theory, Iprobably wouldn’t have made it past page 6.

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References

[1] Saunders Mac Lane, Categories for the Working Mathematician. 2nd edition, 1998.

[2] Saunders Mac Lane and Ieke Moerdijk, Sheaves in Geometry and Logic: A FirstIntroduction to Topos Theory. 1992.

[3] Erik Palmgren, Semantics of Intuitionistic Propositional Logic. 2009.

[4] Peter T. Johnstone, Sketches of an Elephant: A Topos Theory Compendium. 2002.

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