UoM-Fibration Examples • Syntax of UoM ( p : E → L Ab , 1 , num ) where E = Types and Terms • UoM-Fibrations from the Usual Fibrations Codomain, Suboject and Relations fibration over Set are λ 1 -fibrations with simple products. ◦ Choose Abelian group object, G 13
UoM-Fibration Examples • Syntax of UoM ( p : E → L Ab , 1 , num ) where E = Types and Terms • UoM-Fibrations from the Usual Fibrations Codomain, Suboject and Relations fibration over Set are λ 1 -fibrations with simple products. ◦ Choose Abelian group object, G ◦ Choose object in fibre above G 13
UoM-Fibration Examples • Syntax of UoM ( p : E → L Ab , 1 , num ) where E = Types and Terms • UoM-Fibrations from the Usual Fibrations Codomain, Suboject and Relations fibration over Set are λ 1 -fibrations with simple products. ◦ Choose Abelian group object, G ◦ Choose object in fibre above G • Unit Erasure Semantics ( p : E → 1 , ∗ , num ) e.g. E = cpo and num = Q ⊥ 13
Theorems About UoM-Fibrations 14
Theorems About UoM-Fibrations Theorem • ( p : E → B , G , X ) UoM-fibration • A finite products • F : A → B product preserving functor • G ′ ∈ A an Abelian group object with FG ′ = G 14
Theorems About UoM-Fibrations Theorem • ( p : E → B , G , X ) UoM-fibration • A finite products • F : A → B product preserving functor • G ′ ∈ A an Abelian group object with FG ′ = G Then ( F ∗ p , G ′ , ( G ′ , X )) is a UoM-fibration. 14
Theorems About UoM-Fibrations Theorem • ( p : E → B , G , X ) UoM-fibration F ∗ E ✲ E • A finite products • F : A → B product preserving functor F ∗ p p • G ′ ∈ A an Abelian group object with ❄ ❄ FG ′ = G ✲ B A F Then ( F ∗ p , G ′ , ( G ′ , X )) is a UoM-fibration. 14
Theorems About UoM-Fibrations ctd... 15
Theorems About UoM-Fibrations ctd... Theorem Any UoM-fibration can be converted into a UoM-fibration with L Ab in the base. 15
Theorems About UoM-Fibrations ctd... Theorem Any UoM-fibration can be converted into a UoM-fibration with L Ab in the base. ✲ E F ∗ E F ∗ p p F ( 1 ) = G ❄ ❄ ✲ B L Ab F 15
Another Example Recall an Abelian group G can be thought of as a category G 16
Another Example Recall an Abelian group G can be thought of as a category G • Ob ( G ) = ∗ 16
Another Example Recall an Abelian group G can be thought of as a category G • Ob ( G ) = ∗ • G ( ∗ , ∗ ) = G 16
Another Example Recall an Abelian group G can be thought of as a category G • Ob ( G ) = ∗ • G ( ∗ , ∗ ) = G A G -Set is a functor φ : G → Set, i.e., 16
Another Example Recall an Abelian group G can be thought of as a category G • Ob ( G ) = ∗ • G ( ∗ , ∗ ) = G A G -Set is a functor φ : G → Set, i.e., • φ ∗ ∈ Set, which we denote | φ | 16
Another Example Recall an Abelian group G can be thought of as a category G • Ob ( G ) = ∗ • G ( ∗ , ∗ ) = G A G -Set is a functor φ : G → Set, i.e., • φ ∗ ∈ Set, which we denote | φ | • φ g : | φ | → | φ | . 16
Another Example Recall an Abelian group G can be thought of as a category G • Ob ( G ) = ∗ • G ( ∗ , ∗ ) = G A G -Set is a functor φ : G → Set, i.e., • φ ∗ ∈ Set, which we denote | φ | • φ g : | φ | → | φ | . Definition We call the functor p : Ab-Set → Ab the Ab-Set fibration, where Ab-Set G = [ G , Set ] 16
Another Example Recall an Abelian group G can be thought of as a category G • Ob ( G ) = ∗ • G ( ∗ , ∗ ) = G A G -Set is a functor φ : G → Set, i.e., • φ ∗ ∈ Set, which we denote | φ | • φ g : | φ | → | φ | . Definition We call the functor p : Ab-Set → Ab the Ab-Set fibration, where Ab-Set G = [ G , Set ] Theorem The Ab - Set fibration is a λ 1 -fibration with simple products. Hence, for choices G ∈ Ab , num ∈ Ab - Set G ( p : Ab - Set → Ab , G , num ) is a UoM-fibration 16
Theorem About Fibrations 17
Theorem About Fibrations Theorem Let E and B be categories with finite products. 17
Theorem About Fibrations Theorem Let E and B be categories with finite products. • Suppose that [ ] : B → Cat is a product preserving functor. 17
Theorem About Fibrations Theorem Let E and B be categories with finite products. • Suppose that [ ] : B → Cat is a product preserving functor. • p : E → B is a fibration with E X := [ X ] → D and hence reindexing is given by precomposition 17
Theorem About Fibrations Theorem Let E and B be categories with finite products. • Suppose that [ ] : B → Cat is a product preserving functor. • p : E → B is a fibration with E X := [ X ] → D and hence reindexing is given by precomposition ◦ i.e., for any f : X → Y ∈ B , f ∗ ( φ : [ Y ] → D ) = φ ◦ [ f ] . Then, the reindexing of any projection map π X : X × Y → X has a right adjoint π ∗ X ⊣ Ran [ π ] , which satisfies the Beck-Chevalley condition. 17
Sketch Proof 18
Sketch Proof Lemma For π : X × Y → X in B and φ : [ X ] × [ Y ] → D in E X × Y then 18
Sketch Proof Lemma For π : X × Y → X in B and φ : [ X ] × [ Y ] → D in E X × Y then ( Ran [ π ] φ ) x = lim y ∈ [ Y ] φ ( x , y ) 18
Sketch Proof Keep in mind: ( Ran [ π ] φ ) x = lim y ∈ [ Y ] φ ( x , y ) 19
Sketch Proof Keep in mind: ( Ran [ π ] φ ) x = lim y ∈ [ Y ] φ ( x , y ) Want to show: 19
Sketch Proof Keep in mind: ( Ran [ π ] φ ) x = lim y ∈ [ Y ] φ ( x , y ) f : X → X ′ Want to show: For any in B and ψ : [ X ′ ] × [ Y ] → D in E X ′× Y ( Ran π X ( f × id ) ∗ ψ ) x ∼ = ( f ∗ Ran π X ′ ψ ) x 19
Sketch Proof Keep in mind: ( Ran [ π ] φ ) x = lim y ∈ [ Y ] φ ( x , y ) f : X → X ′ Want to show: For any in B and ψ : [ X ′ ] × [ Y ] → D in E X ′× Y ( Ran π X ( f × id ) ∗ ψ ) x ∼ = ( f ∗ Ran π X ′ ψ ) x Use Lemma: 19
Sketch Proof Keep in mind: ( Ran [ π ] φ ) x = lim y ∈ [ Y ] φ ( x , y ) f : X → X ′ Want to show: For any in B and ψ : [ X ′ ] × [ Y ] → D in E X ′× Y ( Ran π X ( f × id ) ∗ ψ ) x ∼ = ( f ∗ Ran π X ′ ψ ) x Use Lemma: ( Ran π X ( f × id ) ∗ ψ ) x ∼ y ∈ Y ( f × id ) ∗ φ ( x , y ) ∼ = lim = lim y ∈ Y φ ( fx , y ) 19
Sketch Proof Keep in mind: ( Ran [ π ] φ ) x = lim y ∈ [ Y ] φ ( x , y ) f : X → X ′ Want to show: For any in B and ψ : [ X ′ ] × [ Y ] → D in E X ′× Y ( Ran π X ( f × id ) ∗ ψ ) x ∼ = ( f ∗ Ran π X ′ ψ ) x Use Lemma: ( Ran π X ( f × id ) ∗ ψ ) x ∼ y ∈ Y ( f × id ) ∗ φ ( x , y ) ∼ = lim = lim y ∈ Y φ ( fx , y ) ( f ∗ Ran π X ′ ψ ) x ∼ = lim y ∈ Y φ ( fx , y ) 19
Summary of Last Few Slides 20
Summary of Last Few Slides • If fibration such that reindexing is given by precomposition 20
Summary of Last Few Slides • If fibration such that reindexing is given by precomposition • AND right adjoints are given by right Kan extensions 20
Summary of Last Few Slides • If fibration such that reindexing is given by precomposition • AND right adjoints are given by right Kan extensions • Then quantification satisfies BC 20
Summary of Last Few Slides • If fibration such that reindexing is given by precomposition • AND right adjoints are given by right Kan extensions • Then quantification satisfies BC • We use this to show the Ab-Set fibration is a UoM-fibration 20
Results in the Ab-Set Fibration 21
Results in the Ab-Set Fibration Lemma u ⊢ S , T Type , Suppose then | � ∀ u . S → T � | ∼ = Nat ( � S � , � T � ) 21
Results in the Ab-Set Fibration Lemma u ⊢ S , T Type , Suppose then | � ∀ u . S → T � | ∼ = Nat ( � S � , � T � ) Proof. By end formula for a Kan extension. 21
Results in the Ab-Set Fibration ctd... Lemma t : ∀ u . num ( u ) → num ( u n ) Let for some m , n ∈ N , 22
Results in the Ab-Set Fibration ctd... Lemma t : ∀ u . num ( u ) → num ( u n ) Let for some m , n ∈ N , then for x ∈ | num ( u ) | � t � ( g · x ) = g n · ( � t � x ) ∀ g ∈ G 22
Results in the Ab-Set Fibration ctd... Lemma t : ∀ u . num ( u ) → num ( u n ) Let for some m , n ∈ N , then for x ∈ | num ( u ) | � t � ( g · x ) = g n · ( � t � x ) ∀ g ∈ G Proof. Use previous lemma to see � t � ∈ G -Set ( num ( u ) , num ( u n )) 22
Results in the Ab-Set Fibration ctd... Lemma t : ∀ u . num ( u ) → num ( u n ) Let for some m , n ∈ N , then for x ∈ | num ( u ) | � t � ( g · x ) = g n · ( � t � x ) ∀ g ∈ G Proof. Use previous lemma to see � t � ∈ G -Set ( num ( u ) , num ( u n )) Naturality gives result. 22
Results in the Ab-Set Fibration ctd... Corollary There is no non-trivial term of type ∀ u . num ( u 2 ) → num ( u ) . 23
Results in the Ab-Set Fibration ctd... Corollary There is no non-trivial term of type ∀ u . num ( u 2 ) → num ( u ) . Proof. Consider ( p : Ab-Set → Ab , Z 2 , Z 2 ) , 23
Results in the Ab-Set Fibration ctd... Corollary There is no non-trivial term of type ∀ u . num ( u 2 ) → num ( u ) . Proof. Consider ( p : Ab-Set → Ab , Z 2 , Z 2 ) , Then if there were a term t : ∀ u . num ( u 2 ) → num ( u ) , then � t � ( g 2 · x ) = g · ( � t � x ) ∀ g ∈ Z 2 23
Results in the Ab-Set Fibration ctd... Corollary There is no non-trivial term of type ∀ u . num ( u 2 ) → num ( u ) . Proof. Consider ( p : Ab-Set → Ab , Z 2 , Z 2 ) , Then if there were a term t : ∀ u . num ( u 2 ) → num ( u ) , then � t � ( g 2 · x ) = g · ( � t � x ) ∀ g ∈ Z 2 Which does not hold, because If � t � 0 = 1 then � t � ( 1 + 1 + 0 ) = 1 + � t � ( 0 ) 23
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