An Elementary Approach to Elementary Topos Theory

Todd Trimble

Western Connecticut State University Department of Mathematics

October 26, 2019 I Standard approach: forbiddingly technical (monadicity criteria, Beck-Chevalley conditions, ...) for those who grew up on naive set theory.

I Tierney’s approach: constructions are more natively ”set-theoretical”.

Back Story

I Tierney’s approach: private communication. I Tierney’s approach: constructions are more natively ”set-theoretical”.

Back Story

I Tierney’s approach: private communication.

I Standard approach: forbiddingly technical (monadicity criteria, Beck-Chevalley conditions, ...) for those who grew up on naive set theory. Back Story

I Tierney’s approach: private communication.

I Standard approach: forbiddingly technical (monadicity criteria, Beck-Chevalley conditions, ...) for those who grew up on naive set theory.

I Tierney’s approach: constructions are more natively ”set-theoretical”. I Construction of coproducts: X + Y is an equalizer:

uP(PX ×PY ) −→ X +Y  P(PX ×PY ) −→ PPP(PX ×PY ) PhP(PπPX ◦uX ),P(PπPY ◦uY )i

u : 1E → PP is unit uX (x) = {A : PX |x ∈ A}

Back Story

I Standard approach to deduce existence of colimits: P : E op → E is monadic. Back Story

I Standard approach to deduce existence of colimits: P : E op → E is monadic.

I Construction of coproducts: X + Y is an equalizer:

uP(PX ×PY ) −→ X +Y  P(PX ×PY ) −→ PPP(PX ×PY ) PhP(PπPX ◦uX ),P(PπPY ◦uY )i

u : 1E → PP is unit uX (x) = {A : PX |x ∈ A} I singX : X → PX classifies δX : X → X × X .

Notation and Preliminaries

I Power-object definition of topos: finite limits, universal relations 3X ,→ PX × X .

R ,→ X × Y

X → PY χR

R 3Y y i χ ×1 X × Y i Y PY × Y Notation and Preliminaries

I Power-object definition of topos: finite limits, universal relations 3X ,→ PX × X .

R ,→ X × Y

X → PY χR

R 3Y y i χ ×1 X × Y i Y PY × Y

I singX : X → PX classifies δX : X → X × X . I Toposes are balanced.

Notation and Preliminaries

I 31= 1 → P1 × 1, aka t : 1 → Ω.

I All monos are regular:

χ A i X i Ω t ! 1

I Epi-mono factorizations are unique when they exist. Notation and Preliminaries

I 31= 1 → P1 × 1, aka t : 1 → Ω.

I All monos are regular:

χ A i X i Ω t ! 1

I Epi-mono factorizations are unique when they exist.

I Toposes are balanced. Cartesian closure

I Exponentials PZ Y exist, namely P(Y × Z) =∼ (PZ)Y :

X → P(Y × Z)

R → X × Y × Z

X × Y → PZ

X → PZ Y

I X 1 X Y 1Y

y y Y singX t t Y PX τ P1 PX Y τ P1Y I Colimits in E/Y , when they exist, are stable under pullback f ∗ : E/Y → E/X .

Slice theorem

I If E is a topos, then for any object X , the category E/X is also a topos. The change of base X ∗ : E → E/X is logical and has left and right adjoints.

I f ∗ : E/Y → (E/Y )/f ' E/X , for f : X → Y , is logical. Slice theorem

I If E is a topos, then for any object X , the category E/X is also a topos. The change of base X ∗ : E → E/X is logical and has left and right adjoints.

I f ∗ : E/Y → (E/Y )/f ' E/X , for f : X → Y , is logical.

I Colimits in E/Y , when they exist, are stable under pullback f ∗ : E/Y → E/X . [≤] ,→ Ω × Ω

⇒ = χ[≤] :Ω × Ω → Ω

Internal logic

1 × 1 t→×t Ω × Ω

∧ = χt×t :Ω × Ω → Ω Internal logic

1 × 1 t→×t Ω × Ω

∧ = χt×t :Ω × Ω → Ω

[≤] ,→ Ω × Ω

⇒ = χ[≤] :Ω × Ω → Ω T Define X : PPX → PX by \ F = {x : X | ∀A:PX A ∈PX F ⇒ x ∈X A}

Internal logic

X →! 1 →t Ω

X tX : 1 → Ω = PX

∀X = χtX : PX → Ω Internal logic

X →! 1 →t Ω

X tX : 1 → Ω = PX

∀X = χtX : PX → Ω

T Define X : PPX → PX by \ F = {x : X | ∀A:PX A ∈PX F ⇒ x ∈X A} Construction of coproducts

I Initial object: define 0 ,→ 1 to be “intersection all subobjects of 1”, classified by

t T 1 →P1 PP1 → P1

I Lemma: 0 is initial.

I Uniqueness: if f , g : 0 ⇒ X , then Eq(f , g)  0 is an equality, by minimality of 0 in Sub(1). I Existence: consider

P X y singX t 0 1 X PX I Given X , Y , disjointly embed them into PX × PY :

χ ×χ χ ×χ X × 1 δ 0 PX × PY 1 × Y 0 δ PX × PY

X t Y is the “disjoint union”: the intersection of the definable family of subobjects of PX × PY containing these embeddings.

Coproducts

I 0 is strict by cartesian closure, so 0 → X is monic. Coproducts

I 0 is strict by cartesian closure, so 0 → X is monic.

I Given X , Y , disjointly embed them into PX × PY :

χ ×χ χ ×χ X × 1 δ 0 PX × PY 1 × Y 0 δ PX × PY

X t Y is the “disjoint union”: the intersection of the definable family of subobjects of PX × PY containing these embeddings. Coproducts

I Lemma: Any two disjoint unions of X , Y are isomorphic.

I Proof: If Z = X ∪ Y via i : X → Z and j : Y → Z, then map Z into PX × PY via

h1 ,ii h1 ,ji X ,→X X × ZY ,→Y Y × Z

Z → PX Z → PY

Then Z → PX × PY is monic. Both Z and X t Y are least upper bounds of X and Y in Sub(PX × PY ).  Coproducts

I Theorem: X t Y is the coproduct.

I Proof: Given f : X → B and g : Y → B, form

h1 ,f i h1 ,gi X ,→X X × B, Y ,→Y Y × B.

Then (X t Y ) × B =∼ (X × B) t (Y × B). So both X , Y embed disjointly in (X t Y ) × B. Obtain

X t Y ,→ (X t Y ) × B.  I B X B y 1X X f Y X f Y

T ∗ I im(f ) = Y {B : PY | f B = X }

Image factorization

I For f : X → Y , define im(f ) to be the intersection of the (definable) family of subobjects through which f factors. T ∗ I im(f ) = Y {B : PY | f B = X }

Image factorization

I For f : X → Y , define im(f ) to be the intersection of the (definable) family of subobjects through which f factors.

I B X B y 1X X f Y X f Y Image factorization

I For f : X → Y , define im(f ) to be the intersection of the (definable) family of subobjects through which f factors.

I B X B y 1X X f Y X f Y

T ∗ I im(f ) = Y {B : PY | f B = X } Image factorization

I Lemma: f : X → Y indeed factors through im(f ): I → Y .

I Proof: We must show f ∗(im(f )) = X . But

! ∗ f ∗ T B = T f ∗B [E/Y →f E/X is logical] B | f ∗B=X B | f ∗B=X

= T X B | f ∗B=X

= X  Image factorization

I Lemma: X → im(f ) ,→ Y is the epi-mono factorization of f : X → Y .

Proof: Put I = im(f ); suppose X → I equalizes g, h : I ⇒ Z. Then

X → Eq(g, h)  I ,→ Y

makes Eq(g, h) a subobject through which f factors. Hence Eq(g, h) = I and g = h.  I Form the image factorization of hf , gi : X → Y × Y :

X → R  Y × Y

I The equivalence relation E on Y generated by R : P(Y × Y ) is the intersection of the definable family of equivalence relations containing R.

I Form the classifying map χE : Y → PY of E ,→ Y × Y (mapping y : Y to its E-equivalence class).

I Form the image factorization of χE :

Y  Q  PY

I Theorem: Y → Q is the coequalizer of f , g.

Coequalizers Let f , g : X → Y be maps. Construct coequalizer X ⇒ Y → Q: I The equivalence relation E on Y generated by R : P(Y × Y ) is the intersection of the definable family of equivalence relations containing R.

I Form the classifying map χE : Y → PY of E ,→ Y × Y (mapping y : Y to its E-equivalence class).

I Form the image factorization of χE :

Y  Q  PY

I Theorem: Y → Q is the coequalizer of f , g.

Coequalizers Let f , g : X → Y be maps. Construct coequalizer X ⇒ Y → Q: I Form the image factorization of hf , gi : X → Y × Y :

X → R  Y × Y I Form the classifying map χE : Y → PY of E ,→ Y × Y (mapping y : Y to its E-equivalence class).

I Form the image factorization of χE :

Y  Q  PY

I Theorem: Y → Q is the coequalizer of f , g.

Coequalizers Let f , g : X → Y be maps. Construct coequalizer X ⇒ Y → Q: I Form the image factorization of hf , gi : X → Y × Y :

X → R  Y × Y

I The equivalence relation E on Y generated by R : P(Y × Y ) is the intersection of the definable family of equivalence relations containing R. I Form the image factorization of χE :

Y  Q  PY

I Theorem: Y → Q is the coequalizer of f , g.

Coequalizers Let f , g : X → Y be maps. Construct coequalizer X ⇒ Y → Q: I Form the image factorization of hf , gi : X → Y × Y :

X → R  Y × Y

I The equivalence relation E on Y generated by R : P(Y × Y ) is the intersection of the definable family of equivalence relations containing R.

I Form the classifying map χE : Y → PY of E ,→ Y × Y (mapping y : Y to its E-equivalence class). I Theorem: Y → Q is the coequalizer of f , g.

Coequalizers Let f , g : X → Y be maps. Construct coequalizer X ⇒ Y → Q: I Form the image factorization of hf , gi : X → Y × Y :

X → R  Y × Y

I The equivalence relation E on Y generated by R : P(Y × Y ) is the intersection of the definable family of equivalence relations containing R.

I Form the classifying map χE : Y → PY of E ,→ Y × Y (mapping y : Y to its E-equivalence class).

I Form the image factorization of χE :

Y  Q  PY Coequalizers Let f , g : X → Y be maps. Construct coequalizer X ⇒ Y → Q: I Form the image factorization of hf , gi : X → Y × Y :

X → R  Y × Y

I The equivalence relation E on Y generated by R : P(Y × Y ) is the intersection of the definable family of equivalence relations containing R.

I Form the classifying map χE : Y → PY of E ,→ Y × Y (mapping y : Y to its E-equivalence class).

I Form the image factorization of χE :

Y  Q  PY

I Theorem: Y → Q is the coequalizer of f , g.