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 on naive set theory.

Back Story ◮ Tierney’s approach: private communication. ◮ Standard approach: forbiddingly technical (monadicity criteria, Beck-Chevalley conditions, ...) for those who grew up on 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: uP ( PX × PY ) − → X + Y ֌ P ( PX × PY ) P � P ( P π PX ◦ u X ) , P ( P π PY ◦ u Y ) � PPP ( PX × PY ) − → u : 1 E → PP is unit u X ( x ) = { A : PX | x ∈ A }

Notation and Preliminaries ◮ Power-object definition of topos: finite limits, universal relations ∋ X ֒ → PX × X . R ֒ → X × Y X → χ R PY ∋ Y R � i χ i × 1 Y X × Y PY × Y

Notation and Preliminaries ◮ Power-object definition of topos: finite limits, universal relations ∋ X ֒ → PX × X . R ֒ → X × Y X → χ R PY ∋ Y R � i χ i × 1 Y X × Y PY × Y ◮ sing X : X → PX classifies δ X : X → X × X .

Notation and Preliminaries ◮ ∋ 1 = 1 → P 1 × 1, aka t : 1 → Ω. ◮ All monos are regular: i χ i A X Ω t ! 1 ◮ Epi-mono factorizations are unique when they exist.

Notation and Preliminaries ◮ ∋ 1 = 1 → P 1 × 1, aka t : 1 → Ω. ◮ All monos are regular: i χ i A X Ω t ! 1 ◮ 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 Y 1 Y X 1 � � t Y sing X t τ Y τ PX Y P 1 Y PX P 1

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 ∧ = χ t × t : Ω × Ω → Ω

Internal logic 1 × 1 t × t → Ω × Ω ∧ = χ t × t : Ω × Ω → Ω [ ≤ ] ֒ → Ω × Ω ⇒ = χ [ ≤ ] : Ω × Ω → Ω

Internal logic ! t → 1 → Ω X t X : 1 → Ω X = PX ∀ X = χ t X : PX → Ω

Internal logic ! t → 1 → Ω X t X : 1 → Ω X = PX ∀ X = χ t X : 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 of 1”, classified by � t P 1 1 → PP 1 → P 1 ◮ 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 � sing X t X 0 1 PX

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 : χ δ × χ 0 χ 0 × χ δ X × 1 PX × PY 1 × Y PX × PY 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 � 1 X , i � � 1 Y , j � ֒ → X × Z ֒ → Y × Z X Y 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 � 1 X , f � � 1 Y , g � X ֒ → X × B , Y ֒ → 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 � 1 X f f X Y X Y

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 � 1 X f f X Y X Y ◮ 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 � � [ E / Y f ∗ f ∗ f ∗ B � B = � → E / X is logical] B | f ∗ B = X B | f ∗ B = X � = X B | f ∗ B = 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 χ E : 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 χ E : Y → PY of E ֒ → Y × Y (mapping y : Y to its E -equivalence class). ◮ Form the image factorization of χ E : 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 χ E : Y → PY of E ֒ → Y × Y (mapping y : Y to its E -equivalence class). ◮ Form the image factorization of χ E : Y ։ Q ֌ PY ◮ Theorem: Y → Q is the coequalizer of f , g .