Categories of Filters as Fibered Completions Toshiki Kataoka - PowerPoint PPT Presentation

Categories of Filters 
 as Fibered Completions Toshiki Kataoka 2016.08.08

Butz '04, “Saturated models of intuitionistic theories” • Filter logics • 픹 ↪ F 픹 
 → F B B � satisfies a saturation principle Kataoka (UTokyo) 2

Models with 
 saturation principles (classical) • ultrafilter construction � Set Set U � S U S • filter construction [Pitts '83][Palmgren '97] (intuitionistic) F B Sh ( F B ) B � B ( − ) B • [Butz '04] F B B Kataoka (UTokyo) 3

Blass '74, “Two closed categories of filters” • Filt ( 픹 ) , F 픹 Kataoka (UTokyo) 4

• Filt ( 픹 ) , F 픹 Question Why F 픹 has good properties? Why not Filt ( 픹 ) ? Kataoka (UTokyo) 5

Answer categorical models fibrational models ⊆ E ↓ B B Sub ( B ) Filt ( B ) → F B ↓ ↓ B � → � B B [Butz '04] • F 픹 ≅ Filt ( 픹 )[ W -1 ] (localization) Kataoka (UTokyo) 6

Overview 0. Introduction 1. Categories of filters [Koubek & Reiterman '70][Blass '74] 2. Categorical logic for filters [Butz '04] 3. Categorical models vs. fibrational models Kataoka (UTokyo) 7

Overview 0. Introduction 1. Categories of filters [Koubek & Reiterman '70][Blass '74] 2. Categorical logic for filters [Butz '04] 3. Categorical models vs. fibrational models Kataoka (UTokyo) 8

Filters of a semilattice • L : (bounded) (meet-)semilattice Definition F : filter of L def. F ⊆ L : upward closed subset s.t. ⇐ ⇒ • ⊤ ∈ F { • x ∧ y ∈ F if x ∈ F and y ∈ F def. F ≤ G ――― F ⊇ G ⇐ ⇒ Kataoka (UTokyo) 9

Definition F L := { F : filter of L } 
 ≅ SLat ( L , 2) op cat. of cat. of complete semilattices semilattices Theorem F � SLat - SLat � F L L M M Kataoka (UTokyo) 10

Filters on an object • 픹 : category with pullbacks Definition F : filter on I ∈ 픹 def. F : filter of Sub 픹 ( I ) ⇐ ⇒ Kataoka (UTokyo) 11

Definition Filt 픹 ( I ) := { F : filter on I } 
 = F ( Sub 픹 ( I )) Filt B F Sub B B op � SLat - SLat � Kataoka (UTokyo) 12

Two categories of filters [Blass, '74] Filt ( 픹 ) • The category of concrete filters F 픹 • The category of abstract filters Kataoka (UTokyo) 13

Cat. of concrete filters Filt ( 픹 ) • object ( I , F ) ( I ∈ 픹 , F : filter on I ) • morphism in Filt ( B ) u : ( I, F ) → ( J, G ) in B u : I → J ∀ Y ∈ G . u − 1 Y ∈ F Kataoka (UTokyo) 14

Set Filt ( Set ) [Blass '74] Lemma Filt ( B ) ↓ is the Grothendieck construction B from the functor Filt B : B op → � - SLat . Kataoka (UTokyo) 15

Cat. of abstract filters F 픹 • object ( I , F ) ( I ∈ 픹 , F : filter on I ) in F B • morphism [ v ]: ( I, F ) � ( J, G ) is defined as v F � X in B � J � Y � G . v � 1 Y � F (under v − 1 Y ⊆ X ⊆ I ) under def. � v | X �� = v � | X �� ( X �� � X � X � ) [ v ] = [ v � ] � Kataoka (UTokyo) 16

F ( Set ) [Blass '74] Kataoka (UTokyo) 17

Definition F = Colim : B → Set � X ∈ F B ( X, − ) is called the reduced product [K. ?] Lemma op : fully faithful ∏ (_) : F 픹 → ( Set 픹 ) Kataoka (UTokyo) 18

Overview 0. Introduction 1. Categories of filters [Koubek & Reiterman '70][Blass '74] 2. Categorical logic for filters [Butz '04] 3. Categorical models vs. fibrational models Kataoka (UTokyo) 19

First-order logic and 
 its fragments terms t ::= x | f ( t 1 , . . . , t | f | ) ϕ ::= R ( t 1 , . . . , t | R | ) | t 1 = t 2 formulas | � | ϕ 1 � ϕ 2 “left exact logic” | � x. ϕ regular logic | � | ϕ 1 � ϕ 2 coherent logic | ϕ 1 � ϕ 2 | � x. ϕ Kataoka (UTokyo) 20

Categorical models • 픹 : category with finite products • interpretation in 픹 • types ⟦ σ ⟧ ∈ 픹 • function symbols ⟦ f ⟧ ∈ 픹 ( ⟦ σ ⟧ 1 × … × ⟦ σ ⟧ |f| ) • relation symbols ⟦ R ⟧ ∈ Sub 픹 ( ⟦ σ ⟧ 1 × … × ⟦ σ ⟧ |R| ) Kataoka (UTokyo) 21

⇒ ⇒ ⇒ ⇒ 
 
 • 픹 : left exact category 
 픹 models left exact logics 
 (has finite limits) • 픹 : regular category 
 픹 models regular logics 
 (lex. & has p.b.-stable 
 coequalizers of kernel pairs) • 픹 : coherent category 
 픹 models coherent logics 
 (reg. & has 
 p.b.-stable finite unions) • 픹 : Heyting category 픹 models first-order 
 logics Kataoka (UTokyo) 22

Filter logics [Butz '04] • F 픹 models filter logics t ::= x | f ( t 1 , . . . , t | f | ) ϕ ::= R ( t 1 , . . . , t | R | ) | t 1 = t 2 � | � | ϕ 1 � ϕ 2 | ϕ “left exact filter logic” ϕ ∈ Φ | � x. ϕ regular filter logic | � | ϕ 1 � ϕ 2 coherent filter logic | ϕ 1 � ϕ 2 | � x. ϕ Kataoka (UTokyo) 23

Filter logics [Butz '04] for each ϕ � Φ ψ � ϕ � ψ � � ϕ ∈ Φ ϕ � ϕ 0 ϕ ∈ Φ ϕ ψ � � x. ϕ 1 � · · · � ϕ n for each { ϕ 1 , . . . , ϕ n } � Φ fin. ψ � � x. � ϕ ∈ Φ ϕ saturation � ϕ ∈ Φ ( ψ � ϕ ) � ψ � � ϕ ∈ Φ ϕ Kataoka (UTokyo) 24

Characterization of F 픹 [Blass '74] Lemma Sub F 픹 ( I , F ) ≅ { ( I , G ) | G ≤ F } : complete (meet-)semilattice [Butz '04] Definition 픸 : filtered meet lex category def. 픸 : lex category, ⇐ ⇒ Sub 픸 : 픸 op → ⋀ - SLat Kataoka (UTokyo) 25

[Butz '04] Definition 픸 : filtered meet lex category def. 픸 : lex category, ⇐ ⇒ Sub 픸 : 픸 op → ⋀ - SLat [Butz '04] Theorem Lex : category of 
 F � filt Lex - Lex � lex categories filt - Lex : category of ⋀ F B B filtered meet A A lex categories Kataoka (UTokyo) 26

Filtered meet vs. 
 arbitrary meet arbitrary meets = finite meets + filtered meets � � = ( x 1 ∧ · · · ∧ x n ) x x ∈ X { x 1 ,...,x n } ⊆ X fin. Kataoka (UTokyo) 27

� � � � � x. ϕ ( x ) � ψ ( x ) � � x. ϕ ( x ) � x. ψ ( x ) � ✓ ✗ ✗ arbitrary meets = finite meets + filtered meets � � = ( x 1 ∧ · · · ∧ x n ) x x ∈ X { x 1 ,...,x n } ⊆ X fin. Kataoka (UTokyo) 28

[Butz '04] Definition a filtered meet regular (resp. coherent ) category 
 is a regular (resp. coherent) category with filtered meets s.t. ∃ (and ∨ ) distributes over filtered meets [Butz '04] Theorem F � filt Lex - Lex restricts to � F � filt F � filt Reg - Reg Coh - Coh and � � Kataoka (UTokyo) 29

Filter logics [Butz '04] for each ϕ � Φ ψ � ϕ � ψ � � ϕ ∈ Φ ϕ � ϕ 0 ϕ ∈ Φ ϕ ψ � � x. ϕ 1 � · · · � ϕ n for each { ϕ 1 , . . . , ϕ n } � Φ fin. ψ � � x. � ϕ ∈ Φ ϕ � ϕ ∈ Φ ( ψ � ϕ ) � ψ � � ϕ ∈ Φ ϕ distributive laws Kataoka (UTokyo) 30

Overview 0. Introduction 1. Categories of filters [Koubek & Reiterman '70][Blass '74] 2. Categorical logic for filters [Butz '04] 3. Categorical models vs. fibrational models Kataoka (UTokyo) 31

Fibrational models • | : fibration, 픹 has finite products E ↓ p B • ⟦ R ⟧ ∈ 피 ⟦ σ ⟧ 1 × … × ⟦ σ ⟧ |R| • Generalization of categorical model 픹 (subobject model) Sub ( B ) for B : lex category ↓ B Sub ( B ) � σ 1 � × ··· × � σ | R | � = Sub B ( � σ 1 � × · · · × � σ | R | � ) � � � R � ∈ Kataoka (UTokyo) 32

Fibered completion Filt ( B ) Sub ( B ) • | is the fibered completion of ↓ ↓ B B Filt B recall F Sub B B op � SLat - SLat � Kataoka (UTokyo) 33

Given 픹 : categorical model of 
 a (lex/regular/coherent) logic [Butz '04] • F 픹 is the “free” (categorical) model of its filter logic [K.] Filt ( B ) • | is the “free” fibrational model its filter logic ↓ B Kataoka (UTokyo) 34

Fibrations vs. categories (General preicates vs. subobjects) Sub ( B ) Filt ( B ) Sub ( F B ) id F B B B Kataoka (UTokyo) 35

∼ Coproducts over = ( ∃ m ) X X monomorphisms m I J Sub ( B ) has ↓ B } for each monomorphism m in 픹 
 � m � m ∗ satisfying the Beck-Chevalley condition ∃ m : coproduct and the Frobenius reciprocity Kataoka (UTokyo) 36

Coproducts over monomorphisms [K.] Lemma Sub ( B )   � m � m ∗ ↓  K B Sub ( B ) Sub ( A ) � � � �  H : morphism of fibrations 
 ↓ ↓ → B A preserving (1, × , ⊤ , ∧ ) � � K K = Sub ( H ) ⇔ preserves ∃ m for m : mono H Kataoka (UTokyo) 37

Left exact fibrations [K.] Definition A left exact fibration 
 E is a fibered poset s.t. ↓ p B • 픹 has finite limits • p has fibered finite meets • p has coproducts over monomorphisms satisfying Frobenius Kataoka (UTokyo) 38

[K.] Theorem L Lex LexFib � � R E L E [ W − 1 ] ↓ B Sub ( A ) ↓ A A E R ↓ B B where W = { ( X → ( ∃ m ) X | m : I � J, X ∈ E I } Kataoka (UTokyo) 39

Localization of a category • Q W : 피 → 피 [ W -1 ] is universal among 
 F : 피 → 픻 s.t. F ( w ): isom. for w ∈ W Kataoka (UTokyo) 40