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A predicative variant of the effective topos Samuele Maschio - - PowerPoint PPT Presentation

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A predicative variant of the effective topos Samuele Maschio (j.w.w. Maria Emilia Maietti) Dipartimento di Matematica Universit` a di Padova Second Workshop on Mathematical Logic and its Applications Kanazawa, 5-9 march 2018 From the

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A predicative variant of the effective topos

Samuele Maschio (j.w.w. Maria Emilia Maietti)

Dipartimento di Matematica Universit` a di Padova

Second Workshop on Mathematical Logic and its Applications Kanazawa, 5-9 march 2018

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From the Minimalist Foundation to a predicative tripos

From the Minimalist Foundation to a predicative tripos

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From the Minimalist Foundation to a predicative tripos

The Minimalist Foundation (Maietti, Sambin 2005, Maietti 2009) consists

  • f two levels:
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From the Minimalist Foundation to a predicative tripos

The Minimalist Foundation (Maietti, Sambin 2005, Maietti 2009) consists

  • f two levels:

an intensional level (to extract computational contents of math)

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SLIDE 5

From the Minimalist Foundation to a predicative tripos

The Minimalist Foundation (Maietti, Sambin 2005, Maietti 2009) consists

  • f two levels:

an intensional level (to extract computational contents of math) an extensional level (actual mathematics)

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SLIDE 6

From the Minimalist Foundation to a predicative tripos

The Minimalist Foundation (Maietti, Sambin 2005, Maietti 2009) consists

  • f two levels:

an intensional level (to extract computational contents of math) an extensional level (actual mathematics) connected via a setoid interpretation

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SLIDE 7

From the Minimalist Foundation to a predicative tripos

The Minimalist Foundation (Maietti, Sambin 2005, Maietti 2009) consists

  • f two levels:

an intensional level (to extract computational contents of math) an extensional level (actual mathematics) connected via a setoid interpretation both contain four kinds of types: sets, collections, small propositions, propositions (in context):

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SLIDE 8

From the Minimalist Foundation to a predicative tripos

The Minimalist Foundation (Maietti, Sambin 2005, Maietti 2009) consists

  • f two levels:

an intensional level (to extract computational contents of math) an extensional level (actual mathematics) connected via a setoid interpretation both contain four kinds of types: sets, collections, small propositions, propositions (in context): mTT setoid emTT Set Col interpretation Set Col Props

  • Prop
  • ⇐

Props

  • Prop
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SLIDE 9

From the Minimalist Foundation to a predicative tripos

The Minimalist Foundation (Maietti, Sambin 2005, Maietti 2009) consists

  • f two levels:

an intensional level (to extract computational contents of math) an extensional level (actual mathematics) connected via a setoid interpretation both contain four kinds of types: sets, collections, small propositions, propositions (in context): mTT setoid emTT Set Col interpretation Set Col Props

  • Prop
  • ⇐

Props

  • Prop
  • idea: mimic this structure to define a predicative version of the effective

topos from a predicative effective tripos.

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SLIDE 10

From the Minimalist Foundation to a predicative tripos

A model for the intensional level

Concrete starting point: the model for mTT + CT + AC in (Ishihara, Maietti, Maschio, Streicher).

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SLIDE 11

From the Minimalist Foundation to a predicative tripos

A model for the intensional level

Concrete starting point: the model for mTT + CT + AC in (Ishihara, Maietti, Maschio, Streicher). this interpretation is performed in Feferman’s ̂ ID1

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SLIDE 12

From the Minimalist Foundation to a predicative tripos

A model for the intensional level

Concrete starting point: the model for mTT + CT + AC in (Ishihara, Maietti, Maschio, Streicher). this interpretation is performed in Feferman’s ̂ ID1 via a variant of Martin-L¨

  • f type theory (props-as-types)
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SLIDE 13

From the Minimalist Foundation to a predicative tripos

A model for the intensional level

Concrete starting point: the model for mTT + CT + AC in (Ishihara, Maietti, Maschio, Streicher). this interpretation is performed in Feferman’s ̂ ID1 via a variant of Martin-L¨

  • f type theory (props-as-types)

not a categorical model: problems with interpretation of λ-abstraction and substitution (weak exponentials)

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From the Minimalist Foundation to a predicative tripos

A model for the intensional level

Concrete starting point: the model for mTT + CT + AC in (Ishihara, Maietti, Maschio, Streicher). this interpretation is performed in Feferman’s ̂ ID1 via a variant of Martin-L¨

  • f type theory (props-as-types)

not a categorical model: problems with interpretation of λ-abstraction and substitution (weak exponentials) however one can extract some categorical structure giving rise to a predicative version of a tripos

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From the Minimalist Foundation to a predicative tripos

A model for the intensional level

Concrete starting point: the model for mTT + CT + AC in (Ishihara, Maietti, Maschio, Streicher). this interpretation is performed in Feferman’s ̂ ID1 via a variant of Martin-L¨

  • f type theory (props-as-types)

not a categorical model: problems with interpretation of λ-abstraction and substitution (weak exponentials) however one can extract some categorical structure giving rise to a predicative version of a tripos

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The predicative effective tripos

The predicative effective tripos

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The predicative effective tripos

The base category

Cr:

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The predicative effective tripos

The base category

Cr:

  • bj A ∶= {x∣ ϕA(x)}, ϕA(x) formula of ̂

ID1 as usual x ε A is ϕA(x);

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SLIDE 19

The predicative effective tripos

The base category

Cr:

  • bj A ∶= {x∣ ϕA(x)}, ϕA(x) formula of ̂

ID1 as usual x ε A is ϕA(x); arr [n]≈A,B ∶ A → B

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The predicative effective tripos

The base category

Cr:

  • bj A ∶= {x∣ ϕA(x)}, ϕA(x) formula of ̂

ID1 as usual x ε A is ϕA(x); arr [n]≈A,B ∶ A → B n numeral;

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The predicative effective tripos

The base category

Cr:

  • bj A ∶= {x∣ ϕA(x)}, ϕA(x) formula of ̂

ID1 as usual x ε A is ϕA(x); arr [n]≈A,B ∶ A → B n numeral; x ε A ⊢ ̂

ID1 {n}(x) ε B;

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The predicative effective tripos

The base category

Cr:

  • bj A ∶= {x∣ ϕA(x)}, ϕA(x) formula of ̂

ID1 as usual x ε A is ϕA(x); arr [n]≈A,B ∶ A → B n numeral; x ε A ⊢ ̂

ID1 {n}(x) ε B;

n ≈A,B m is x ε A ⊢ ̂

ID1 {n}(x) = {m}(x).

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The predicative effective tripos

The base category

Cr:

  • bj A ∶= {x∣ ϕA(x)}, ϕA(x) formula of ̂

ID1 as usual x ε A is ϕA(x); arr [n]≈A,B ∶ A → B n numeral; x ε A ⊢ ̂

ID1 {n}(x) ε B;

n ≈A,B m is x ε A ⊢ ̂

ID1 {n}(x) = {m}(x).

Cr is a finitely complete weakly locally cartesian closed category with parameterized list objects.

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The predicative effective tripos

Collections over Cr

Define over Cr an indexed category representing dependent collections Colr ∶ Cop

r

→ Cat

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The predicative effective tripos

Collections over Cr

Define over Cr an indexed category representing dependent collections Colr ∶ Cop

r

→ Cat Colr(A):

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SLIDE 26

The predicative effective tripos

Collections over Cr

Define over Cr an indexed category representing dependent collections Colr ∶ Cop

r

→ Cat Colr(A):

  • bjects: definable classes with a parameter over the context A
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The predicative effective tripos

Collections over Cr

Define over Cr an indexed category representing dependent collections Colr ∶ Cop

r

→ Cat Colr(A):

  • bjects: definable classes with a parameter over the context A

arrows: recursive functions (possibly depending on the context) represented by numerals

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The predicative effective tripos

Collections over Cr

Define over Cr an indexed category representing dependent collections Colr ∶ Cop

r

→ Cat Colr(A):

  • bjects: definable classes with a parameter over the context A

arrows: recursive functions (possibly depending on the context) represented by numerals Colr([n]) are substitution functors.

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The predicative effective tripos

Collections over Cr

Define over Cr an indexed category representing dependent collections Colr ∶ Cop

r

→ Cat Colr(A):

  • bjects: definable classes with a parameter over the context A

arrows: recursive functions (possibly depending on the context) represented by numerals Colr([n]) are substitution functors.

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The predicative effective tripos

Propositions over Cr

Define over Cr a first-order hyperdoctrine Propr ∶ Cop

r

→ Heyt

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The predicative effective tripos

Propositions over Cr

Define over Cr a first-order hyperdoctrine Propr ∶ Cop

r

→ Heyt the posetal reflection of the doctrine of Kleene realizability for which

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The predicative effective tripos

Propositions over Cr

Define over Cr a first-order hyperdoctrine Propr ∶ Cop

r

→ Heyt the posetal reflection of the doctrine of Kleene realizability for which realized propositions over A are formulas P(x,y) with at most x,y free (we write y ⊩ P(x) instead of P(x,y)) for which x ⊩ P(y) ⊢ ̂

ID1 x εA

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The predicative effective tripos

Propositions over Cr

Define over Cr a first-order hyperdoctrine Propr ∶ Cop

r

→ Heyt the posetal reflection of the doctrine of Kleene realizability for which realized propositions over A are formulas P(x,y) with at most x,y free (we write y ⊩ P(x) instead of P(x,y)) for which x ⊩ P(y) ⊢ ̂

ID1 x εA

P(x,y) ≤ Q(x,y) over A if there exists a numeral r for which y ⊩ P(x) ⊢ ̂

ID1 {r}(x,y) ⊩ Q(x)

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The predicative effective tripos

Propositions over Cr

Define over Cr a first-order hyperdoctrine Propr ∶ Cop

r

→ Heyt the posetal reflection of the doctrine of Kleene realizability for which realized propositions over A are formulas P(x,y) with at most x,y free (we write y ⊩ P(x) instead of P(x,y)) for which x ⊩ P(y) ⊢ ̂

ID1 x εA

P(x,y) ≤ Q(x,y) over A if there exists a numeral r for which y ⊩ P(x) ⊢ ̂

ID1 {r}(x,y) ⊩ Q(x)

different from the tripos defining the effective topos, where x is not a natural number in general y ⊩ P(x) ⊢ ̂

ID1 {r}(y) ⊩ Q(x)

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The predicative effective tripos

Sets and small propositions

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The predicative effective tripos

Sets and small propositions

Last thing to do: there are notions of sets and small propositions to capture.

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The predicative effective tripos

Sets and small propositions

Last thing to do: there are notions of sets and small propositions to capture. define predicates set(x), xǫy and x / ǫ y in ̂ ID1 to encode Martin-L¨

  • f sets

(closed under empty set, singleton, +, Σ, Π, List, Id) and their realizability interpretation;

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SLIDE 38

The predicative effective tripos

Sets and small propositions

Last thing to do: there are notions of sets and small propositions to capture. define predicates set(x), xǫy and x / ǫ y in ̂ ID1 to encode Martin-L¨

  • f sets

(closed under empty set, singleton, +, Σ, Π, List, Id) and their realizability interpretation; define a universe of sets Us in the base category;

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The predicative effective tripos

Sets and small propositions

Last thing to do: there are notions of sets and small propositions to capture. define predicates set(x), xǫy and x / ǫ y in ̂ ID1 to encode Martin-L¨

  • f sets

(closed under empty set, singleton, +, Σ, Π, List, Id) and their realizability interpretation; define a universe of sets Us in the base category; define an indexed category Setr

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SLIDE 40

The predicative effective tripos

Sets and small propositions

Last thing to do: there are notions of sets and small propositions to capture. define predicates set(x), xǫy and x / ǫ y in ̂ ID1 to encode Martin-L¨

  • f sets

(closed under empty set, singleton, +, Σ, Π, List, Id) and their realizability interpretation; define a universe of sets Us in the base category; define an indexed category Setr and small propositions Props

r as a doctrine over the base category.

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The predicative effective topos

The predicative effective topos

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The predicative effective topos

Elementary quotient completion

If p ∶ Cop → Heyt is a first-order hyperdoctrine over a cartesian category C,

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The predicative effective topos

Elementary quotient completion

If p ∶ Cop → Heyt is a first-order hyperdoctrine over a cartesian category C,

  • ne can perform the elementary quotient completion (Maietti, Rosolini)
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The predicative effective topos

Elementary quotient completion

If p ∶ Cop → Heyt is a first-order hyperdoctrine over a cartesian category C,

  • ne can perform the elementary quotient completion (Maietti, Rosolini)

p ∶ Qp → Heyt having all quotients of p-equivalence relations:

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The predicative effective topos

Elementary quotient completion

If p ∶ Cop → Heyt is a first-order hyperdoctrine over a cartesian category C,

  • ne can perform the elementary quotient completion (Maietti, Rosolini)

p ∶ Qp → Heyt having all quotients of p-equivalence relations:

  • bjects of Qp: A = (∣A∣,∼A),

∣A∣ is an object of C and ∼A∈ p(∣A∣ × ∣A∣) is an p-equivalence relation on ∣A∣

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The predicative effective topos

Elementary quotient completion

If p ∶ Cop → Heyt is a first-order hyperdoctrine over a cartesian category C,

  • ne can perform the elementary quotient completion (Maietti, Rosolini)

p ∶ Qp → Heyt having all quotients of p-equivalence relations:

  • bjects of Qp: A = (∣A∣,∼A),

∣A∣ is an object of C and ∼A∈ p(∣A∣ × ∣A∣) is an p-equivalence relation on ∣A∣ arrow in Qp from A to B equivalence class of arrows f ∶ ∣A∣ → ∣B∣ of C respecting equivalence relations: x ∼A y ⊢p f (x) ∼B f (y), w.r.t. to equivalence relation:f ≡ g iff xεA ⊢p f (x) ∼B g(x).

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SLIDE 47

The predicative effective topos

Elementary quotient completion

If p ∶ Cop → Heyt is a first-order hyperdoctrine over a cartesian category C,

  • ne can perform the elementary quotient completion (Maietti, Rosolini)

p ∶ Qp → Heyt having all quotients of p-equivalence relations:

  • bjects of Qp: A = (∣A∣,∼A),

∣A∣ is an object of C and ∼A∈ p(∣A∣ × ∣A∣) is an p-equivalence relation on ∣A∣ arrow in Qp from A to B equivalence class of arrows f ∶ ∣A∣ → ∣B∣ of C respecting equivalence relations: x ∼A y ⊢p f (x) ∼B f (y), w.r.t. to equivalence relation:f ≡ g iff xεA ⊢p f (x) ∼B g(x). α ∈ p(A) iff α ∈ p(∣A∣) and α(x) ∧ x ∼A y ⊢p α(y)

  • rder is inherited from p(∣A∣).
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The predicative effective topos

The predicative effective topos

pEff ∶= QPropr

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The predicative effective topos

The predicative effective topos

pEff ∶= QPropr pEffProp ∶= Propr

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SLIDE 50

The predicative effective topos

The predicative effective topos

pEff ∶= QPropr pEffProp ∶= Propr α ∈ pEffProps(A,∼A) iff α ∈ pEffProp(A,∼A) ∩ Propr

s(∣A∣)

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The predicative effective topos

The predicative effective topos

pEff ∶= QPropr pEffProp ∶= Propr α ∈ pEffProps(A,∼A) iff α ∈ pEffProp(A,∼A) ∩ Propr

s(∣A∣)

  • ne can also define fibrations of collections (codomain fibration)

and of sets over pEff .

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The predicative effective topos

The predicative effective topos

pEff ∶= QPropr pEffProp ∶= Propr α ∈ pEffProps(A,∼A) iff α ∈ pEffProp(A,∼A) ∩ Propr

s(∣A∣)

  • ne can also define fibrations of collections (codomain fibration)

and of sets over pEff .

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The predicative effective topos

Properties of pEff

pEff is a locally cartesian closed list-arithmetic pretopos

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The predicative effective topos

Properties of pEff

pEff is a locally cartesian closed list-arithmetic pretopos it has a classifier Ω for pEffProps, i.e. pEffProps(−) ≃ pEff (−,Ω)

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The predicative effective topos

Properties of pEff

pEff is a locally cartesian closed list-arithmetic pretopos it has a classifier Ω for pEffProps, i.e. pEffProps(−) ≃ pEff (−,Ω) But Propr ≡ wSubCr

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The predicative effective topos

Properties of pEff

pEff is a locally cartesian closed list-arithmetic pretopos it has a classifier Ω for pEffProps, i.e. pEffProps(−) ≃ pEff (−,Ω) But Propr ≡ wSubCr ⇓ pEffProp ≡ SubpEff

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SLIDE 57

The predicative effective topos

Properties of pEff

pEff is a locally cartesian closed list-arithmetic pretopos it has a classifier Ω for pEffProps, i.e. pEffProps(−) ≃ pEff (−,Ω) But Propr ≡ wSubCr ⇓ pEffProp ≡ SubpEff This is a predicative variant of a topos.

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SLIDE 58

The predicative effective topos

Properties of pEff

pEff is a locally cartesian closed list-arithmetic pretopos it has a classifier Ω for pEffProps, i.e. pEffProps(−) ≃ pEff (−,Ω) But Propr ≡ wSubCr ⇓ pEffProp ≡ SubpEff This is a predicative variant of a topos.

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Relation with the Effective Topos

Relation with the Effective Topos

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Relation with the Effective Topos

The effective topos

In ZFC

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Relation with the Effective Topos

The effective topos

In ZFC Assemblies (Asm):

  • bjects: (A,P), A set, P ∶ A → P(N), P(a) ≠ ∅ for every a ∈ A.

arrows f ∶ (A,P) → (B,Q), f ∶ A → B and there is r ∈ N such that {r}(n) ∈ Q(f (a)) for every a ∈ A and n ∈ P(a).

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SLIDE 62

Relation with the Effective Topos

The effective topos

In ZFC Assemblies (Asm):

  • bjects: (A,P), A set, P ∶ A → P(N), P(a) ≠ ∅ for every a ∈ A.

arrows f ∶ (A,P) → (B,Q), f ∶ A → B and there is r ∈ N such that {r}(n) ∈ Q(f (a)) for every a ∈ A and n ∈ P(a). Partitioned Assemblies (pAsm): full subcategory of Asm

  • bjects: #P(a) = 1 for every a ∈ A
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SLIDE 63

Relation with the Effective Topos

The effective topos

In ZFC Assemblies (Asm):

  • bjects: (A,P), A set, P ∶ A → P(N), P(a) ≠ ∅ for every a ∈ A.

arrows f ∶ (A,P) → (B,Q), f ∶ A → B and there is r ∈ N such that {r}(n) ∈ Q(f (a)) for every a ∈ A and n ∈ P(a). Partitioned Assemblies (pAsm): full subcategory of Asm

  • bjects: #P(a) = 1 for every a ∈ A

Rec: full subcategory of Asm

  • bjects: A ⊆ N and P(a) = {a} for every a ∈ A.
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SLIDE 64

Relation with the Effective Topos

The effective topos

In ZFC Assemblies (Asm):

  • bjects: (A,P), A set, P ∶ A → P(N), P(a) ≠ ∅ for every a ∈ A.

arrows f ∶ (A,P) → (B,Q), f ∶ A → B and there is r ∈ N such that {r}(n) ∈ Q(f (a)) for every a ∈ A and n ∈ P(a). Partitioned Assemblies (pAsm): full subcategory of Asm

  • bjects: #P(a) = 1 for every a ∈ A

Rec: full subcategory of Asm

  • bjects: A ⊆ N and P(a) = {a} for every a ∈ A.

Rec isomorphic to the category of subsets of natural numbers and (restrictions of) recursive functions between them.

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SLIDE 65

Relation with the Effective Topos

The effective topos Eff can be introduced in three different ways:

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SLIDE 66

Relation with the Effective Topos

The effective topos Eff can be introduced in three different ways:

1

via tripos-to-topos construction (Hyland, Johnstone, Pitts) from pEff ∶ Setop → preHeyt; with Set from ZFC;

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SLIDE 67

Relation with the Effective Topos

The effective topos Eff can be introduced in three different ways:

1

via tripos-to-topos construction (Hyland, Johnstone, Pitts) from pEff ∶ Setop → preHeyt; with Set from ZFC;

2

as ex/reg completion (Freyd, Carboni, Scedrov)

  • f a category of assemblies Asm;
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SLIDE 68

Relation with the Effective Topos

The effective topos Eff can be introduced in three different ways:

1

via tripos-to-topos construction (Hyland, Johnstone, Pitts) from pEff ∶ Setop → preHeyt; with Set from ZFC;

2

as ex/reg completion (Freyd, Carboni, Scedrov)

  • f a category of assemblies Asm;

3

as ex/lex completion (Robinson, Rosolini)

  • f a category of partitioned assemblies pAsm.
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SLIDE 69

Relation with the Effective Topos

The effective topos Eff can be introduced in three different ways:

1

via tripos-to-topos construction (Hyland, Johnstone, Pitts) from pEff ∶ Setop → preHeyt; with Set from ZFC;

2

as ex/reg completion (Freyd, Carboni, Scedrov)

  • f a category of assemblies Asm;

3

as ex/lex completion (Robinson, Rosolini)

  • f a category of partitioned assemblies pAsm.

Cr is a rendering of Rec in ̂ ID1 and Propr is equivalent to wSub:

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SLIDE 70

Relation with the Effective Topos

The effective topos Eff can be introduced in three different ways:

1

via tripos-to-topos construction (Hyland, Johnstone, Pitts) from pEff ∶ Setop → preHeyt; with Set from ZFC;

2

as ex/reg completion (Freyd, Carboni, Scedrov)

  • f a category of assemblies Asm;

3

as ex/lex completion (Robinson, Rosolini)

  • f a category of partitioned assemblies pAsm.

Cr is a rendering of Rec in ̂ ID1 and Propr is equivalent to wSub: in this case elementary quotient completion is equivalent to ex/lex completion.

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SLIDE 71

Relation with the Effective Topos

The effective topos Eff can be introduced in three different ways:

1

via tripos-to-topos construction (Hyland, Johnstone, Pitts) from pEff ∶ Setop → preHeyt; with Set from ZFC;

2

as ex/reg completion (Freyd, Carboni, Scedrov)

  • f a category of assemblies Asm;

3

as ex/lex completion (Robinson, Rosolini)

  • f a category of partitioned assemblies pAsm.

Cr is a rendering of Rec in ̂ ID1 and Propr is equivalent to wSub: in this case elementary quotient completion is equivalent to ex/lex completion. Hence our construction is a version of 3 restricted to Rec

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SLIDE 72

Relation with the Effective Topos

The effective topos Eff can be introduced in three different ways:

1

via tripos-to-topos construction (Hyland, Johnstone, Pitts) from pEff ∶ Setop → preHeyt; with Set from ZFC;

2

as ex/reg completion (Freyd, Carboni, Scedrov)

  • f a category of assemblies Asm;

3

as ex/lex completion (Robinson, Rosolini)

  • f a category of partitioned assemblies pAsm.

Cr is a rendering of Rec in ̂ ID1 and Propr is equivalent to wSub: in this case elementary quotient completion is equivalent to ex/lex completion. Hence our construction is a version of 3 restricted to Rec (weak subobjects in Rec coincide with weak subobjects in pAsm)

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SLIDE 73

Relation with the Effective Topos

The effective topos Eff can be introduced in three different ways:

1

via tripos-to-topos construction (Hyland, Johnstone, Pitts) from pEff ∶ Setop → preHeyt; with Set from ZFC;

2

as ex/reg completion (Freyd, Carboni, Scedrov)

  • f a category of assemblies Asm;

3

as ex/lex completion (Robinson, Rosolini)

  • f a category of partitioned assemblies pAsm.

Cr is a rendering of Rec in ̂ ID1 and Propr is equivalent to wSub: in this case elementary quotient completion is equivalent to ex/lex completion. Hence our construction is a version of 3 restricted to Rec (weak subobjects in Rec coincide with weak subobjects in pAsm) expressed in ̂ ID1.

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SLIDE 74

Relation with the Effective Topos

References

1

  • M. Hyland, P. Johnstone, A. Pitts. Tripos theory, 1980.

2

  • M. Hyland. The Effective Topos, 1981.

3

  • M. E. Maietti, G. Sambin. Towards a minimalist foundation for constructive

mathematics, 2005.

4

  • M. E. Maietti. A minimalist two-level foundation for constructive

mathematics, 2009.

5

  • M. E. Maietti, G. Rosolini. Quotient completion for the foundation of

constructive mathematics, 2013.

6

H.Ishihara, M.E.Maietti, S. Maschio, T.Streicher. Consistency of the intensional level of the Minimalist Foundation with Church’s Thesis and Axiom of Choice, to appear in Archive for Mathematical Logic.

7

  • M. E. Maietti, S. Maschio. A strictly predicative variant of Hyland’s Effective

Topos, submitted.