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

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Categories of Filters as Fibered Completions Toshiki Kataoka 2016.08.08 Butz '04, Saturated models of intuitionistic theories Filter logics F F B B satisfies a saturation principle Kataoka (UTokyo) 2

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SLIDE 1

Categories of Filters
 as Fibered Completions

Toshiki Kataoka 2016.08.08

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SLIDE 2

Kataoka (UTokyo)

Butz '04, “Saturated models of intuitionistic theories”

  • Filter logics
  • 픹 ↪ F픹


satisfies a saturation principle

2

B → FB

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SLIDE 3

Kataoka (UTokyo)

Models with
 saturation principles

  • ultrafilter construction
  • filter construction [Pitts '83][Palmgren '97]
  • [Butz '04]

3

Set Set S

  • U S

B FB Sh(FB) B

  • (−) B

B FB

  • U

(classical) (intuitionistic)

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SLIDE 4

Kataoka (UTokyo)

Blass '74, “Two closed categories of filters”

  • Filt(픹), F픹

4

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SLIDE 5

Kataoka (UTokyo) 5

  • Filt(픹), F픹

Why F픹 has good properties? Why not Filt(픹)?

Question

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SLIDE 6

Kataoka (UTokyo)

Answer

6

categorical models ⊆ fibrational models B

E ↓ B

B → FB

Sub(B) ↓ B

  • →

Filt(B) ↓ B

  • F픹 ≅ Filt(픹)[W-1] (localization)

[Butz '04]

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SLIDE 7

Kataoka (UTokyo)

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

7

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SLIDE 8

Kataoka (UTokyo)

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

8

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SLIDE 9

Kataoka (UTokyo)

Filters of a semilattice

  • L: (bounded) (meet-)semilattice

9

F : filter of L F ⊆ L: upward closed subset s.t.

  • ⊤ ∈ F
  • x∧y ∈ F if x ∈ F and y ∈ F

F ≤ G ―――F ⊇ G

def.

⇐ ⇒

Definition

def.

⇐ ⇒

{

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SLIDE 10

Kataoka (UTokyo) 10

Theorem

SLat

  • SLat

L FL M M

F

  • cat. of

semilattices

  • cat. of

complete semilattices

FL := { F : filter of L }
 ≅ SLat(L, 2)op

Definition

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SLIDE 11

Kataoka (UTokyo)

Filters on an object

  • 픹: category with pullbacks

11

F : filter on I ∈ 픹 F : filter of Sub픹(I)

def.

⇐ ⇒

Definition

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SLIDE 12

Kataoka (UTokyo) 12

Filt픹(I) := { F : filter on I }
 = F(Sub픹(I))

Definition

Bop SLat

  • SLat

FiltB SubB F

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SLIDE 13

Kataoka (UTokyo)

Two categories of filters

[Blass, '74]

  • The category of concrete filters
  • The category of abstract filters

13

Filt(픹) F픹

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SLIDE 14

Kataoka (UTokyo)

  • Cat. of concrete filters

Filt(픹)

  • object (I, F ) (I ∈ 픹, F : filter on I)
  • morphism

14

u: (I, F) → (J, G) in Filt(B) u: I → J in B ∀Y ∈ G. u−1Y ∈ F

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Kataoka (UTokyo) 15

Lemma

Filt(B) ↓ B

is the Grothendieck construction from the functor FiltB : Bop →

  • SLat.

[Blass '74]

Filt(Set) Set

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SLIDE 16

Kataoka (UTokyo)

  • Cat. of abstract filters

F픹

  • object (I, F ) (I ∈ 픹, F : filter on I)
  • morphism

16

[v]: (I, F) (J, G) in FB is defined as F X

v

J in B Y G. v1Y F

(under v−1Y ⊆ X ⊆ I)

under [v] = [v]

def.

  • v|X = v|X (X X X)
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Kataoka (UTokyo) 17

[Blass '74]

F(Set)

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Kataoka (UTokyo) 18

∏(_): F픹 → (Set픹)

  • p: fully faithful

Lemma

is called the reduced product

Definition

  • F = Colim

X∈F B(X, −)

: B → Set

[K. ?]

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SLIDE 19

Kataoka (UTokyo)

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

19

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SLIDE 20

Kataoka (UTokyo)

First-order logic and
 its fragments

20

t ::= x | f(t1, . . . , t|f|) ϕ ::= R(t1, . . . , t|R|) | t1 = t2 | | ϕ1 ϕ2 | x. ϕ | | ϕ1 ϕ2 | ϕ1 ϕ2 | x. ϕ terms formulas “left exact logic” regular logic coherent logic

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SLIDE 21

Kataoka (UTokyo)

Categorical models

  • 픹: category with finite products
  • interpretation in 픹
  • types ⟦σ⟧ ∈ 픹
  • function symbols ⟦f⟧ ∈ 픹(⟦σ⟧1 × … × ⟦σ⟧|f|)
  • relation symbols ⟦R⟧ ∈ Sub픹(⟦σ⟧1 × … × ⟦σ⟧|R|)

21

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Kataoka (UTokyo)

  • 픹: left exact category


(has finite limits)

  • 픹: regular category


(lex. & has p.b.-stable
 coequalizers of kernel pairs)

  • 픹: coherent category


(reg. & has
 p.b.-stable finite unions)

  • 픹: Heyting category

22

픹 models left exact logics
 픹 models regular logics





픹 models coherent logics





픹 models first-order
 logics ⇒ ⇒ ⇒ ⇒

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SLIDE 23

Kataoka (UTokyo)

Filter logics [Butz '04]

23

“left exact filter logic” regular filter logic coherent filter logic t ::= x | f(t1, . . . , t|f|) ϕ ::= R(t1, . . . , t|R|) | t1 = t2 | | ϕ1 ϕ2 |

  • ϕ∈Φ

ϕ | x. ϕ | | ϕ1 ϕ2 | ϕ1 ϕ2 | x. ϕ

  • F픹 models filter logics
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SLIDE 24

Kataoka (UTokyo)

Filter logics [Butz '04]

24

  • ϕ∈Φ ϕ ϕ0

ψ ϕ for each ϕ Φ ψ

ϕ∈Φ ϕ

ψ x. ϕ1 · · · ϕn for each {ϕ1, . . . , ϕn}

fin.

Φ ψ x.

ϕ∈Φ ϕ

  • ϕ∈Φ(ψ ϕ) ψ

ϕ∈Φ ϕ

saturation

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Kataoka (UTokyo)

Characterization of F픹

25

SubF픹(I, F ) ≅ { (I, G ) | G ≤ F } : complete (meet-)semilattice

Lemma [Blass '74]

픸: filtered meet lex category 픸: lex category, Sub픸: 픸op → ⋀

  • SLat

Definition [Butz '04]

def.

⇐ ⇒

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Kataoka (UTokyo) 26

Lex: category of
 lex categories

⋀

filt-Lex: category of

filtered meet lex categories

Theorem [Butz '04]

픸: filtered meet lex category 픸: lex category, Sub픸: 픸op → ⋀

  • SLat

Definition [Butz '04]

def.

⇐ ⇒ Lex filt

  • Lex

B FB A A

F

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SLIDE 27

Kataoka (UTokyo)

Filtered meet vs.
 arbitrary meet

27

arbitrary meets = finite meets + filtered meets

  • x∈X

x =

  • {x1,...,xn}⊆

fin.

X

(x1 ∧ · · · ∧ xn)

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SLIDE 28

Kataoka (UTokyo) 28

arbitrary meets = finite meets + filtered meets

  • x∈X

x =

  • {x1,...,xn}⊆

fin.

X

(x1 ∧ · · · ∧ xn)

✗ ✓ ✗

x.ϕ(x) ψ(x)

  • x.ϕ(x)
  • x.ψ(x)
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SLIDE 29

Kataoka (UTokyo) 29

Theorem [Butz '04]

a filtered meet regular (resp. coherent) category
 is a regular (resp. coherent) category with filtered meets s.t. ∃ (and ∨) distributes over filtered meets

Definition [Butz '04]

Lex filt

  • Lex

Reg filt

  • Reg

Coh filt

  • Coh

F

  • F
  • F
  • restricts to

and

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Kataoka (UTokyo)

Filter logics [Butz '04]

30

  • ϕ∈Φ ϕ ϕ0

ψ ϕ for each ϕ Φ ψ

ϕ∈Φ ϕ

ψ x. ϕ1 · · · ϕn for each {ϕ1, . . . , ϕn}

fin.

Φ ψ x.

ϕ∈Φ ϕ

  • ϕ∈Φ(ψ ϕ) ψ

ϕ∈Φ ϕ

distributive laws

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SLIDE 31

Kataoka (UTokyo)

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

31

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Kataoka (UTokyo)

Fibrational models

  • | : fibration, 픹 has finite products
  • ⟦R⟧ ∈ 피⟦σ⟧1 × … × ⟦σ⟧|R|
  • Generalization of categorical model 픹

32

E ↓p B Sub(B) ↓ B

for B: lex category R ∈

  • Sub(B)
  • σ1×···×σ|R| = SubB(σ1 × · · · × σ|R|)

(subobject model)

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SLIDE 33

Kataoka (UTokyo)

Fibered completion

  • | is the fibered completion of

33

Filt(B) ↓ B Sub(B) ↓ B

Bop SLat

  • SLat

FiltB SubB F

  • recall
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Kataoka (UTokyo)

Given 픹: categorical model of
 a (lex/regular/coherent) logic

  • F픹 is the “free” (categorical) model of its filter logic
  • | is the “free” fibrational model its filter logic

34

Filt(B) ↓ B

[Butz '04] [K.]

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Kataoka (UTokyo)

Fibrations vs. categories

35

(General preicates vs. subobjects)

Sub(B) Filt(B) Sub(FB) B B FB

id

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Kataoka (UTokyo)

Coproducts over monomorphisms

36

Sub(B) ↓ B

m m∗ X (∃m)X I J

∼ = m

}

: coproduct

∃m has for each monomorphism m in 픹
 satisfying the Beck-Chevalley condition and the Frobenius reciprocity

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SLIDE 37

Kataoka (UTokyo)

Coproducts over monomorphisms

37

Sub(B) ↓ B

m m∗ : morphism of fibrations 
 preserving (1, ×, ⊤, ∧) K = Sub(H) ⇔

Lemma [K.]

  • Sub(B)

↓ B

K H  

→

  • Sub(A)

↓ A

  • K

H

  • preserves ∃m for m: mono
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SLIDE 38

Kataoka (UTokyo)

Left exact fibrations

38

A left exact fibration
 is a fibered poset s.t.

  • 픹 has finite limits
  • p has fibered finite meets
  • p has coproducts over monomorphisms

satisfying Frobenius

Definition [K.]

E ↓p B

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SLIDE 39

Kataoka (UTokyo) 39

Theorem [K.]

Lex LexFib E[W −1]

E ↓ B

A

Sub(A) ↓ A

B

E ↓ B

L R L R

where W = {(X → (∃m)X | m: I J, X ∈ EI}

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Kataoka (UTokyo)

Localization of a category

  • QW: 피 → 피[W-1] is universal among 


F: 피 → 픻 s.t. F(w): isom. for w ∈ W

40

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Kataoka (UTokyo)

Proof of the theorem (1)

41

H : A B: lex

  • K

H

  • :
  • Sub(A)

↓ A

  • E

↓ B

  • : mor. of lex fibrations

K(A) := (i)A for A SubA(A) where i: A A

Lex LexFib E[W −1]

E ↓ B

A

Sub(A) ↓ A

B

E ↓ B

L R L R

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SLIDE 42

Kataoka (UTokyo)

Lex LexFib E[W −1]

E ↓ B

A

Sub(A) ↓ A

B

E ↓ B

L R L R

Proof of the theorem (2)

42

  • K

H

  • :
  • E

↓ B

  • →
  • Sub(A)

↓ A

  • : mor. of lex fibrations

H : E[W 1] → A: lex M := (E

K

→ Sub(A)

dom

→ A) M induces M by the universality

cod

  • For w ∈ W, M'w is isom. (because


Kw is opCartesian lifting, too)

where W = {(X → (∃m)X | m: I J, X ∈ EI}

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Kataoka (UTokyo)

Proof of the theorem (3)

43

H : E[W 1] A: lex

  • K

H

  • :
  • E

↓ B

  • Sub(A)

↓ A

  • : mor. of lex fibrations

M := (E

QW

E[W 1]

M

A) H(I) := M (I) for I B K(X) := M X SubB(H(I)) for X EI

where W = {(X → (∃m)X | m: I J, X ∈ EI}

Lex LexFib E[W −1]

E ↓ B

A

Sub(A) ↓ A

B

E ↓ B

L R L R

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Kataoka (UTokyo) 44

LexFib filt

  • LexFib

Lex filt

  • Lex

R

  • L
  • F
  • [Butz '04]
  • Thm. [K.]
  • Cor. [K.]

fibered
 completion

  • Sub(B)

↓ B

  • Filt(B)

↓ B

  • B

FB

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Kataoka (UTokyo)

Technical condition (*)

45

Condition (*) The embedding is an isomorphism

[K.] Lemma

EI SubE[W −1](I)

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Kataoka (UTokyo) 46

Theorem [K.]

restricts to defined on filtered meet lex fibrations satisfying (*) Lex LexFib filt

  • Lex

filt

  • LexFib
  • L
  • L

LexFib filt

  • LexFib

Lex filt

  • Lex

R

  • L
  • F
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SLIDE 47

Kataoka (UTokyo) 47 LexFib filt

  • LexFib

Lex filt

  • Lex

R

  • L
  • F
  • Define regular fibrations and coherent fibrations


as usual (e.g. [Jacobs '99])
 but we require base categories 픹 have finite limits

(not only finite products)

L restricts to [⋀

filt-](Reg/Coh)Fib [⋀ filt-](Reg/Coh)


defined on fibrations satisfying (*) R restricts to (Reg/Coh)Fib (Reg/Coh)
 defined on fibrations satisfying (*)

Theorem [K.]

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SLIDE 48

Kataoka (UTokyo)

Related work (1)

  • FRel(p): category of functional relations


FRel: RegFib → Reg

  • objects X ∈ 피 i.e. a pair (I ∈ 픹, X ∈ 피I)
  • morphisms R ∈ 피I×J s.t. “functional relation”
  • The tripos-topos construction [Pitts '81]
  • objects (I ∈ 픹, X ∈ 피I×I : “partial equivalence”)

48

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SLIDE 49

Kataoka (UTokyo)

Related work (2)

  • Logical characterization of subobject fibrations
  • Eq-fibration with strong equality
  • has full subset types
  • has unique-choice

49

(e.g. [Jacobs '99])

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SLIDE 50

Kataoka (UTokyo)

Summary

50

FB ∼ = L

  • Filt(B)

↓ B

  • , which is a localization of Filt(B)

categorical models fibrational models B → FB

Sub(B) ↓ B

  • →

Filt(B) ↓ B

L R

[Butz '04]