Relative Partial Combinatory Algebras over Heyting Categories Jetze - - PowerPoint PPT Presentation

Relative Partial Combinatory Algebras over Heyting Categories

Jetze Zoethout Category Theory, 8 July 2019

Jetze Zoethout Relative PCAs over Heyting Categories CT2019 1 / 21

Table of Contents

1

Background and Motivation

2

PCAs over Heyting Categories

3

Slicing

4

Computational Density

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Table of Contents

1

Background and Motivation

2

PCAs over Heyting Categories

3

Slicing

4

Computational Density

Jetze Zoethout Relative PCAs over Heyting Categories CT2019 3 / 21

Partial Combinatory Algebras

Definition

A partial combinatory algebra (PCA) is a nonempty set A with a partial binary operation A × A ⇀ A: (a, b) → ab for which there exist k, s ∈ A such that:

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Partial Combinatory Algebras

Definition

A partial combinatory algebra (PCA) is a nonempty set A with a partial binary operation A × A ⇀ A: (a, b) → ab for which there exist k, s ∈ A such that: (i) kab = a; (here abc = (ab)c) (ii) sab is always defined; (iii) if ac(bc) is defined, then sabc is defined and equal to ac(bc).

Jetze Zoethout Relative PCAs over Heyting Categories CT2019 4 / 21

Partial Combinatory Algebras

Definition

A partial combinatory algebra (PCA) is a nonempty set A with a partial binary operation A × A ⇀ A: (a, b) → ab for which there exist k, s ∈ A such that: (i) kab = a; (here abc = (ab)c) (ii) sab is always defined; (iii) if ac(bc) is defined, then sabc is defined and equal to ac(bc).

Property

If t( x, y) is a term, then there exists an r ∈ A such that:

Jetze Zoethout Relative PCAs over Heyting Categories CT2019 4 / 21

Partial Combinatory Algebras

Definition

A partial combinatory algebra (PCA) is a nonempty set A with a partial binary operation A × A ⇀ A: (a, b) → ab for which there exist k, s ∈ A such that: (i) kab = a; (here abc = (ab)c) (ii) sab is always defined; (iii) if ac(bc) is defined, then sabc is defined and equal to ac(bc).

Property

If t( x, y) is a term, then there exists an r ∈ A such that: (i) r a is defined; (ii) if t( a, b) is defined, then r ab is defined and equal to t( a, b).

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Relative PCAs

Definition

A relative PCA is a pair (A, C) where A is a PCA, and C ⊆ A closed under the application from A, such that there exist k, s ∈ C witnessing the fact that A is a PCA.

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Relative PCAs

Definition

A relative PCA is a pair (A, C) where A is a PCA, and C ⊆ A closed under the application from A, such that there exist k, s ∈ C witnessing the fact that A is a PCA. We view the elements of C as computable elements acting on possibly non-computable data.

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Examples of PCAs

Example

Kleene’s first model K1 is N with mn = ϕm(n).

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Examples of PCAs

Example

Kleene’s first model K1 is N with mn = ϕm(n).

Example

Scott’s graph model is a total PCA with underlying set PN, such that a function (PN)n → PN is computable if and only if it is Scott continuous.

Jetze Zoethout Relative PCAs over Heyting Categories CT2019 6 / 21

Examples of PCAs

Example

Kleene’s first model K1 is N with mn = ϕm(n).

Example

Scott’s graph model is a total PCA with underlying set PN, such that a function (PN)n → PN is computable if and only if it is Scott continuous. (PN, (PN)r.e.) is a relative PCA.

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Assemblies

Definition

The category Asm(A, C): (i) has as objects pairs X = (|X|, EX), where |X| is a set and EX ⊆ |X| × A satisfies: for all x ∈ |X|, there is an a ∈ A with EX(x, a).

Jetze Zoethout Relative PCAs over Heyting Categories CT2019 7 / 21

Assemblies

Definition

The category Asm(A, C): (i) has as objects pairs X = (|X|, EX), where |X| is a set and EX ⊆ |X| × A satisfies: for all x ∈ |X|, there is an a ∈ A with EX(x, a). (ii) arrows X → Y are functions |X| → |Y | for which there exists a tracker r ∈ C such that: if EX(x, a), then ra is defined and EY (f (x), ra).

Jetze Zoethout Relative PCAs over Heyting Categories CT2019 7 / 21

Assemblies

Definition

The category Asm(A, C): (i) has as objects pairs X = (|X|, EX), where |X| is a set and EX ⊆ |X| × A satisfies: for all x ∈ |X|, there is an a ∈ A with EX(x, a). (ii) arrows X → Y are functions |X| → |Y | for which there exists a tracker r ∈ C such that: if EX(x, a), then ra is defined and EY (f (x), ra). The category Asm(A) is a quasitopos.

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Slices of Realizability Categories

Question

What does a category of the form Asm(A)/I or Asm(A, C)/I look like?

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Slices of Realizability Categories

Question

What does a category of the form Asm(A)/I or Asm(A, C)/I look like?

• 1. Are these slice categories again realizability categories of some kind?

Jetze Zoethout Relative PCAs over Heyting Categories CT2019 8 / 21

Slices of Realizability Categories

Question

What does a category of the form Asm(A)/I or Asm(A, C)/I look like?

• 1. Are these slice categories again realizability categories of some kind?
• 2. Can we find a convenient description of these slice categories?

Jetze Zoethout Relative PCAs over Heyting Categories CT2019 8 / 21

Slices of Realizability Categories

Question

What does a category of the form Asm(A)/I or Asm(A, C)/I look like?

• 1. Are these slice categories again realizability categories of some kind?
• 2. Can we find a convenient description of these slice categories?

There is an adjunction Set Asm(A, C)

∇ Γ

with Γ ⊣ ∇.

Jetze Zoethout Relative PCAs over Heyting Categories CT2019 8 / 21

Table of Contents

1

Background and Motivation

2

PCAs over Heyting Categories

3

Slicing

4

Computational Density

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HPCAs

Let H be a locally small Heyting category.

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HPCAs

Let H be a locally small Heyting category.

Definition (Stekelenburg)

An HPCA over H is a pair (A, φ), where A is an inhabited object of H with a binary partial map A × A ⇀ A and φ (the filter) is a set of inhabited subobjects of A such that:

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HPCAs

Let H be a locally small Heyting category.

Definition (Stekelenburg)

An HPCA over H is a pair (A, φ), where A is an inhabited object of H with a binary partial map A × A ⇀ A and φ (the filter) is a set of inhabited subobjects of A such that: (i) φ is upwards closed; (ii) φ is closed under application;

Jetze Zoethout Relative PCAs over Heyting Categories CT2019 10 / 21

HPCAs

Let H be a locally small Heyting category.

Definition (Stekelenburg)

An HPCA over H is a pair (A, φ), where A is an inhabited object of H with a binary partial map A × A ⇀ A and φ (the filter) is a set of inhabited subobjects of A such that: (i) φ is upwards closed; (ii) φ is closed under application; (iii) for every term t( x, y), there exists a U ∈ φ such that ∀r ∈ U ∀ a ∈ A(r a↓ ∧ ∀b ∈ A(t( a, b)↓ → r ab↓ ∧ (r ab = t( a, b)))). is valid in H.

Jetze Zoethout Relative PCAs over Heyting Categories CT2019 10 / 21

HPCAs

Let H be a locally small Heyting category.

Definition (Stekelenburg)

An HPCA over H is a pair (A, φ), where A is an inhabited object of H with a binary partial map A × A ⇀ A and φ (the filter) is a set of inhabited subobjects of A such that: (i) φ is upwards closed; (ii) φ is closed under application; (iii) for every term t( x, y), there exists a U ∈ φ such that ∀r ∈ U ∀ a ∈ A(r a↓ ∧ ∀b ∈ A(t( a, b)↓ → r ab↓ ∧ (r ab = t( a, b)))). is valid in H. There is also a notion of morphism between HPCAs over H.

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The Category HPCA

Proposition (Z)

If (A, φ) is an HPCA over H and p : H → G is a Heyting functor, then p∗(A, φ) := (p(A), p(φ)) is an HPCA over G;

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The Category HPCA

Proposition (Z)

If (A, φ) is an HPCA over H and p : H → G is a Heyting functor, then p∗(A, φ) := (p(A), p(φ)) is an HPCA over G; and this assignment is functorial in both (A, φ) and H.

Jetze Zoethout Relative PCAs over Heyting Categories CT2019 11 / 21

The Category HPCA

Proposition (Z)

If (A, φ) is an HPCA over H and p : H → G is a Heyting functor, then p∗(A, φ) := (p(A), p(φ)) is an HPCA over G; and this assignment is functorial in both (A, φ) and H. We get a category HPCA:

Jetze Zoethout Relative PCAs over Heyting Categories CT2019 11 / 21

The Category HPCA

Proposition (Z)

If (A, φ) is an HPCA over H and p : H → G is a Heyting functor, then p∗(A, φ) := (p(A), p(φ)) is an HPCA over G; and this assignment is functorial in both (A, φ) and H. We get a category HPCA: (A, φ) p∗(A, φ) (B, ψ) H G

f p

Jetze Zoethout Relative PCAs over Heyting Categories CT2019 11 / 21

The Category HPCA

Proposition (Z)

If (A, φ) is an HPCA over H and p : H → G is a Heyting functor, then p∗(A, φ) := (p(A), p(φ)) is an HPCA over G; and this assignment is functorial in both (A, φ) and H. We get a category HPCA: (A, φ) p∗(A, φ) (B, ψ) H G

f p

The pair (p, f ) is called an applicative morphism.

Jetze Zoethout Relative PCAs over Heyting Categories CT2019 11 / 21

The Category HPCA

Proposition (Z)

If (A, φ) is an HPCA over H and p : H → G is a Heyting functor, then p∗(A, φ) := (p(A), p(φ)) is an HPCA over G; and this assignment is functorial in both (A, φ) and H. We get a category HPCA: (A, φ) p∗(A, φ) (B, ψ) H G

f p

The pair (p, f ) is called an applicative morphism. The category HPCA has small products.

Jetze Zoethout Relative PCAs over Heyting Categories CT2019 11 / 21

Assemblies Again

Definition

The category Asm(A, φ): (i) has as objects pairs X = (|X|, EX), where |X| ∈ H and EX ⊆ |X| × A is such that ∀x ∈ |X|∃a ∈ A(EX(x, a)) is valid in H.

Jetze Zoethout Relative PCAs over Heyting Categories CT2019 12 / 21

Assemblies Again

Definition

The category Asm(A, φ): (i) has as objects pairs X = (|X|, EX), where |X| ∈ H and EX ⊆ |X| × A is such that ∀x ∈ |X|∃a ∈ A(EX(x, a)) is valid in H. (ii) arrows X → Y are arrows |X| → |Y | of H for which there exists U ∈ φ such that: ∀r ∈ U ∀x ∈ |X|∀a ∈ A(EX(x, a) → (ra↓ ∧ EY (f (x), ra))) is valid in H.

Jetze Zoethout Relative PCAs over Heyting Categories CT2019 12 / 21

Assemblies Again

Definition

The category Asm(A, φ): (i) has as objects pairs X = (|X|, EX), where |X| ∈ H and EX ⊆ |X| × A is such that ∀x ∈ |X|∃a ∈ A(EX(x, a)) is valid in H. (ii) arrows X → Y are arrows |X| → |Y | of H for which there exists U ∈ φ such that: ∀r ∈ U ∀x ∈ |X|∀a ∈ A(EX(x, a) → (ra↓ ∧ EY (f (x), ra))) is valid in H. An applicative morphism (p, f ): (A, φ) → (B, ψ) also induces a functor Asm(p, f ): Asm(A, φ) → Asm(B, ψ).

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Assemblies Again

Definition

The category Asm(A, φ): (i) has as objects pairs X = (|X|, EX), where |X| ∈ H and EX ⊆ |X| × A is such that ∀x ∈ |X|∃a ∈ A(EX(x, a)) is valid in H. (ii) arrows X → Y are arrows |X| → |Y | of H for which there exists U ∈ φ such that: ∀r ∈ U ∀x ∈ |X|∀a ∈ A(EX(x, a) → (ra↓ ∧ EY (f (x), ra))) is valid in H. An applicative morphism (p, f ): (A, φ) → (B, ψ) also induces a functor Asm(p, f ): Asm(A, φ) → Asm(B, ψ). The functor Asm preserves small products.

Jetze Zoethout Relative PCAs over Heyting Categories CT2019 12 / 21

Table of Contents

1

Background and Motivation

2

PCAs over Heyting Categories

3

Slicing

4

Computational Density

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Description of Slice Categories

Theorem (Stekelenburg)

Categories of the form Asm(A, φ) are closed under slicing.

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Description of Slice Categories

Theorem (Stekelenburg)

Categories of the form Asm(A, φ) are closed under slicing. Let I ∈ Asm(A, φ), and consider the Heyting functor |I|∗ : H → H/|I|.

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Description of Slice Categories

Theorem (Stekelenburg)

Categories of the form Asm(A, φ) are closed under slicing. Let I ∈ Asm(A, φ), and consider the Heyting functor |I|∗ : H → H/|I|.

Theorem (Z)

Asm(A, φ)/I is equivalent to Asm((A, φ)/I), where (A, φ)/I := (|I|∗(A), |I|∗(φ) ∪ {EI}). (Observe that EI ⊆ |I| × A = |I|∗(A).)

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Examples of Slice Categories I

Example

Consider 1 + 1 ∈ Asm(K1). Then K1/1 + 1 ∼ = ((K1, K1), φmax), so Asm(K1)/1 + 1 ≃ Asm(K1)2.

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Examples of Slice Categories I

Example

Consider 1 + 1 ∈ Asm(K1). Then K1/1 + 1 ∼ = ((K1, K1), φmax), so Asm(K1)/1 + 1 ≃ Asm(K1)2.

Example

Consider ∇2 ∈ Asm(K1). Then K1/∇2 ∼ = ((K1, K1), φ), where φ = {(U0, U1) ⊆ (N, N) | U0 ∩ U1 = ∅} = φmax.

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Examples of Slice Categories I

Example

Consider 1 + 1 ∈ Asm(K1). Then K1/1 + 1 ∼ = ((K1, K1), φmax), so Asm(K1)/1 + 1 ≃ Asm(K1)2.

Example

Consider ∇2 ∈ Asm(K1). Then K1/∇2 ∼ = ((K1, K1), φ), where φ = {(U0, U1) ⊆ (N, N) | U0 ∩ U1 = ∅} = φmax. An arrow (f0, f1): (X0, X1) → (Y0, Y1) of Asm(K1)2 belongs to Asm((K1, K1), φ) if f0 and f1 have a simultaneous tracker.

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Examples of Slice Categories II

Example

Consider Σ ∈ Asm(K1) where |Σ| = 2 and EΣ = {(0, n) | nn↓} ∪ {(1, n) | nn↑}. Then K1/Σ = ((K1, K1), φ) is not generated by some C ⊆ (N, N).

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Examples of Slice Categories II

Example

Consider Σ ∈ Asm(K1) where |Σ| = 2 and EΣ = {(0, n) | nn↓} ∪ {(1, n) | nn↑}. Then K1/Σ = ((K1, K1), φ) is not generated by some C ⊆ (N, N). An arrow (f0, f1): (X0, X1) → (Y0, Y1) of Asm(K1)2 belongs to Asm((K1, K1), φ) if, for some total recursive function g, we have that g(n) tracks f0 if nn↓, while g(n) tracks f1 if nn↑.

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Examples of Slice Categories III

Example

The natural numbers object N ∈ Asm(K1) is given by |N| = N and EN = δ ⊆ N × N. We have K1/N ∼ = ((K1)n∈N, φ), where φ = {(Un)n∈N | ∃a ∈ NN

rec∀n ∈ N(an ∈ Un)}.

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Examples of Slice Categories III

Example

The natural numbers object N ∈ Asm(K1) is given by |N| = N and EN = δ ⊆ N × N. We have K1/N ∼ = ((K1)n∈N, φ), where φ = {(Un)n∈N | ∃a ∈ NN

rec∀n ∈ N(an ∈ Un)}.

Example

The natural numbers object N ∈ Asm(PN) is given by |N| = N and EN = {(n, {n}) | n ∈ N}. We have PN/N ∼ = ((PN)n∈N, φmax) ∼ =

• n∈N

PN,

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Examples of Slice Categories III

Example

The natural numbers object N ∈ Asm(K1) is given by |N| = N and EN = δ ⊆ N × N. We have K1/N ∼ = ((K1)n∈N, φ), where φ = {(Un)n∈N | ∃a ∈ NN

rec∀n ∈ N(an ∈ Un)}.

Example

The natural numbers object N ∈ Asm(PN) is given by |N| = N and EN = {(n, {n}) | n ∈ N}. We have PN/N ∼ = ((PN)n∈N, φmax) ∼ =

• n∈N

PN, so Asm(PN)/N ≃

n∈N Asm(PN).

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Table of Contents

1

Background and Motivation

2

PCAs over Heyting Categories

3

Slicing

4

Computational Density

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Geometric morphisms

Let p : G → H be an open geometric morphism between toposes, and suppose we have an applicative morphism (p∗, f ): (A, φ) → (B, ψ).

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Geometric morphisms

Let p : G → H be an open geometric morphism between toposes, and suppose we have an applicative morphism (p∗, f ): (A, φ) → (B, ψ). Asm(B, ψ) Asm(A, φ) G H

Γ Γ Asm(p∗,f ) p∗ ∇ p∗ ∇

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Geometric morphisms

Let p : G → H be an open geometric morphism between toposes, and suppose we have an applicative morphism (p∗, f ): (A, φ) → (B, ψ). Asm(B, ψ) Asm(A, φ) G H

Γ Γ Asm(p∗,f ) p∗ ∇ p∗ ∇

Question

When doen Asm(p∗, f ) have a right adjoint?

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Computational Density

As for PCAs, there exists a notion of computationally dense applicative morphism, which answers this question.

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Computational Density

As for PCAs, there exists a notion of computationally dense applicative morphism, which answers this question.

Example

If (A, φ) is an HPCA over H and p : H → G is a Heyting functor, then the cocartesian arrow (A, φ) → p∗(A, φ) is computationally dense.

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Computational Density

As for PCAs, there exists a notion of computationally dense applicative morphism, which answers this question.

Example

If (A, φ) is an HPCA over H and p : H → G is a Heyting functor, then the cocartesian arrow (A, φ) → p∗(A, φ) is computationally dense. In particular, the projections

j∈J(Aj, φj) → (Aj, φj) are computationally

dense.

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Computational Density

As for PCAs, there exists a notion of computationally dense applicative morphism, which answers this question.

Example

If (A, φ) is an HPCA over H and p : H → G is a Heyting functor, then the cocartesian arrow (A, φ) → p∗(A, φ) is computationally dense. In particular, the projections

j∈J(Aj, φj) → (Aj, φj) are computationally

dense.

Example

If I ∈ Asm(A, φ), then there is a canonical applicative morphism (A, φ) → (A, φ)/I, which is computationally dense.

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References

• J. Frey.

A fibrational study of realizability toposes. PhD thesis, Universit´ e Paris Diderot (Paris 7), 2014.

• P. T. Johnstone.

Geometric morphisms of realizability toposes. Theory and Applications of Categories, 28(9):241–49, 2013.

• W. P. Stekelenburg.

Realizability Categories. PhD thesis, Utrecht University, 2013.

• J. van Oosten.

Realizability: An Introduction to its Categorical Side, volume 152 of Studies in Logic and the Foundations of Mathematics. Elsevier, 2008.

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