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 Background and Motivation 1 PCAs over Heyting Categories 2 Slicing 3 Computational Density 4 Jetze Zoethout Relative PCAs over Heyting Categories CT2019 2 / 21
Table of Contents Background and Motivation 1 PCAs over Heyting Categories 2 Slicing 3 Computational Density 4 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: 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) k ab = a ; (here abc = ( ab ) c ) (ii) s ab is always defined; (iii) if ac ( bc ) is defined, then s abc 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) k ab = a ; (here abc = ( ab ) c ) (ii) s ab is always defined; (iii) if ac ( bc ) is defined, then s abc 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) k ab = a ; (here abc = ( ab ) c ) (ii) s ab is always defined; (iii) if ac ( bc ) is defined, then s abc 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 ). Jetze Zoethout Relative PCAs over Heyting Categories CT2019 4 / 21
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. Jetze Zoethout Relative PCAs over Heyting Categories CT2019 5 / 21
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. Jetze Zoethout Relative PCAs over Heyting Categories CT2019 5 / 21
Examples of PCAs Example Kleene’s first model K 1 is N with mn = ϕ m ( n ). Jetze Zoethout Relative PCAs over Heyting Categories CT2019 6 / 21
Examples of PCAs Example Kleene’s first model K 1 is N with mn = ϕ m ( n ). Example Scott’s graph model is a total PCA with underlying set P N , such that a function ( P N ) n → P N 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 K 1 is N with mn = ϕ m ( n ). Example Scott’s graph model is a total PCA with underlying set P N , such that a function ( P N ) n → P N is computable if and only if it is Scott continuous. ( P N , ( P N ) r.e. ) is a relative PCA. Jetze Zoethout Relative PCAs over Heyting Categories CT2019 6 / 21
Assemblies Definition The category Asm( A , C ): (i) has as objects pairs X = ( | X | , E X ), where | X | is a set and E X ⊆ | X | × A satisfies: for all x ∈ | X | , there is an a ∈ A with E X ( 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 | , E X ), where | X | is a set and E X ⊆ | X | × A satisfies: for all x ∈ | X | , there is an a ∈ A with E X ( x , a ). (ii) arrows X → Y are functions | X | → | Y | for which there exists a tracker r ∈ C such that: if E X ( x , a ), then ra is defined and E Y ( 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 | , E X ), where | X | is a set and E X ⊆ | X | × A satisfies: for all x ∈ | X | , there is an a ∈ A with E X ( x , a ). (ii) arrows X → Y are functions | X | → | Y | for which there exists a tracker r ∈ C such that: if E X ( x , a ), then ra is defined and E Y ( f ( x ) , ra ). The category Asm( A ) is a quasitopos. Jetze Zoethout Relative PCAs over Heyting Categories CT2019 7 / 21
Slices of Realizability Categories Question What does a category of the form Asm( A ) / I or Asm( A , C ) / I look like? 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? 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 Γ Set Asm( A , C ) ∇ with Γ ⊣ ∇ . Jetze Zoethout Relative PCAs over Heyting Categories CT2019 8 / 21
Table of Contents Background and Motivation 1 PCAs over Heyting Categories 2 Slicing 3 Computational Density 4 Jetze Zoethout Relative PCAs over Heyting Categories CT2019 9 / 21
HPCAs Let H be a locally small Heyting category. 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: 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; 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 . Jetze Zoethout Relative PCAs over Heyting Categories CT2019 10 / 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 ; 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 . 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
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