An Elementary Approach to Elementary Topos Theory Todd Trimble - - PowerPoint PPT Presentation

An Elementary Approach to Elementary Topos Theory

Todd Trimble

Western Connecticut State University Department of Mathematics

October 26, 2019

Back Story

◮ Tierney’s approach: private communication.

Back Story

◮ Tierney’s approach: private communication. ◮ Standard approach: forbiddingly technical (monadicity criteria, Beck-Chevalley conditions, ...) for those who grew up

• n naive set theory.

Back Story

◮ Tierney’s approach: private communication. ◮ Standard approach: forbiddingly technical (monadicity criteria, Beck-Chevalley conditions, ...) for those who grew up

• n naive set theory.

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

Back Story

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

Back Story

◮ Standard approach to deduce existence of colimits: P : E op → E is monadic. ◮ Construction of coproducts: X + Y is an equalizer: X+Y ֌ P(PX×PY )

uP(PX×PY )

− →

− →

PP(PπPX ◦uX ),P(PπPY ◦uY ) PPP(PX×PY )

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

Notation and Preliminaries

◮ Power-object definition of topos: finite limits, universal relations ∋X֒ → PX × X. R ֒ → X × Y X →

χR PY

R ∋Y X × Y PY × Y

i

• χi×1Y

Notation and Preliminaries

◮ Power-object definition of topos: finite limits, universal relations ∋X֒ → PX × X. R ֒ → X × Y X →

χR PY

R ∋Y X × Y PY × Y

i

• χi×1Y

◮ singX : X → PX classifies δX : X → X × X.

Notation and Preliminaries

◮ ∋1= 1 → P1 × 1, aka t : 1 → Ω. ◮ All monos are regular: A X Ω 1

i χi ! t

◮ Epi-mono factorizations are unique when they exist.

Notation and Preliminaries

◮ ∋1= 1 → P1 × 1, aka t : 1 → Ω. ◮ All monos are regular: A X Ω 1

i χi ! t

◮ Epi-mono factorizations are unique when they exist. ◮ Toposes are balanced.

Cartesian closure

◮ Exponentials PZ Y exist, namely P(Y × Z) ∼ = (PZ)Y : X → P(Y × Z) R → X × Y × Z X × Y → PZ X → PZ Y ◮ X 1 X Y 1Y PX P1 PX Y P1Y

singX

• t
• tY

τ τ Y

Slice theorem

◮ 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. ◮ f ∗ : E/Y → (E/Y )/f ≃ E/X, for f : X → Y , is logical.

Slice theorem

◮ 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. ◮ f ∗ : E/Y → (E/Y )/f ≃ E/X, for f : X → Y , is logical. ◮ Colimits in E/Y , when they exist, are stable under pullback f ∗ : E/Y → E/X.

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

Internal logic

X

!

→ 1

t

→ Ω tX : 1 → ΩX = PX ∀X = χtX : PX → Ω

Internal logic

X

!

→ 1

t

→ Ω tX : 1 → ΩX = PX ∀X = χtX : PX → Ω Define

X : PPX → PX by

• F = {x : X | ∀A:PX A ∈PX F ⇒ x ∈X A}

Construction of coproducts

◮ Initial object: define 0 ֒ → 1 to be “intersection all subobjects

• f 1”, classified by

1

tP1

→ PP1

• → P1

◮ Lemma: 0 is initial. ◮ Uniqueness: if f , g : 0 ⇒ X, then Eq(f , g) ֌ 0 is an equality, by minimality of 0 in Sub(1). ◮ Existence: consider P X 1 PX

• singX

tX

Coproducts

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

Coproducts

◮ 0 is strict by cartesian closure, so 0 → X is monic. ◮ Given X, Y , disjointly embed them into PX × PY : X × 1 PX × PY 1 × Y PX × PY

χδ×χ0 χ0×χδ

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

Coproducts

◮ Lemma: Any two disjoint unions of X, Y are isomorphic. ◮ Proof: If Z = X ∪ Y via i : X → Z and j : Y → Z, then map Z into PX × PY via X

1X ,i

֒ → X × Z Y

1Y ,j

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

Coproducts

◮ Theorem: X ⊔ Y is the coproduct. ◮ Proof: Given f : X → B and g : Y → B, form X

1X ,f

֒ → X × B, Y

1Y ,g

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

Image factorization

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

Image factorization

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

1X

• f

f

Image factorization

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

1X

• f

f

◮ im(f ) =

Y {B : PY | f ∗B = X}

Image factorization

◮ Lemma: f : X → Y indeed factors through im(f ) : I → Y . ◮ Proof: We must show f ∗(im(f )) = X. But f ∗

• B | f ∗B=X

B

• =
• B | f ∗B=X

f ∗B [E/Y f ∗ → E/X is logical] =

• B | f ∗B=X

X = X

Image factorization

◮ 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.

Coequalizers

Let f , g : X → Y be maps. Construct coequalizer X ⇒ Y → Q:

Coequalizers

Let f , g : X → Y be maps. Construct coequalizer X ⇒ Y → Q: ◮ Form the image factorization of f , g : X → Y × Y : X → R ֌ Y × Y

Coequalizers

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

Coequalizers

Let f , g : X → Y be maps. Construct coequalizer X ⇒ Y → Q: ◮ Form the image factorization of f , g : X → Y × Y : X → R ֌ Y × Y ◮ The equivalence relation E on Y generated by R : P(Y × Y ) is the intersection of the definable family of equivalence relations containing R. ◮ Form the classifying map χE : Y → PY of E ֒ → Y × Y (mapping y : Y to its E-equivalence class).

Coequalizers

Let f , g : X → Y be maps. Construct coequalizer X ⇒ Y → Q: ◮ Form the image factorization of f , g : X → Y × Y : X → R ֌ Y × Y ◮ The equivalence relation E on Y generated by R : P(Y × Y ) is the intersection of the definable family of equivalence relations containing R. ◮ Form the classifying map χE : Y → PY of E ֒ → Y × Y (mapping y : Y to its E-equivalence class). ◮ Form the image factorization of χE: Y ։ Q ֌ PY

Coequalizers

Let f , g : X → Y be maps. Construct coequalizer X ⇒ Y → Q: ◮ Form the image factorization of f , g : X → Y × Y : X → R ֌ Y × Y ◮ The equivalence relation E on Y generated by R : P(Y × Y ) is the intersection of the definable family of equivalence relations containing R. ◮ Form the classifying map χE : Y → PY of E ֒ → Y × Y (mapping y : Y to its E-equivalence class). ◮ Form the image factorization of χE: Y ։ Q ֌ PY ◮ Theorem: Y → Q is the coequalizer of f , g.