Hilberts -operator in categorical logic Fabio Pasquali University - PowerPoint PPT Presentation

Second Workshop on Mathematical Logic and its Applications 8 March 2018 - Kanazawa - Japan Hilbert’s ǫ -operator in categorical logic Fabio Pasquali University of Padova j.w.w. M.E. Maietti (Univ.of Padova) & G. Rosolini (Univ.of Genova)

Primary doctrines C has finite products. A primary doctrine is a functor P : C op → InfSL

� Primary doctrines C has finite products. A primary doctrine is a functor P : C op → InfSL X f Y

� � Primary doctrines C has finite products. A primary doctrine is a functor P : C op → InfSL X P ( X ) f P ( f ) Y P ( Y )

� � Example: contravariant powerset functor P : Set op − → InfSL P ( X ) X f P ( f )= f − 1 P ( Y ) Y

� � Example: contravariant powerset functor P : Set op − → InfSL P ( X ) X f P ( f )= f − 1 P ( Y ) Y P ( A ) is ordered by ⊆ Finite meets are ∩

Elementary and existential doctrines [F.W. Lawvere]

Elementary and existential doctrines P : C op → InfSL is elementary and existential if it has “direct images” [F.W. Lawvere]

Elementary and existential doctrines P : C op → InfSL is elementary and existential if it has “direct images”: i.e. for all f : X → A , there is a covariant ‘natural’ assignment ∃ f : P ( X ) → P ( A ) such that ∃ f ( α ) ≤ β α ≤ P ( f )( β ) [F.W. Lawvere]

Elementary and existential doctrines P : C op → InfSL is elementary and existential if it has “direct images”: i.e. for all f : X → A , there is a covariant ‘natural’ assignment ∃ f : P ( X ) → P ( A ) such that ∃ f ( α ) ≤ β α ≤ P ( f )( β ) Equality: [F.W. Lawvere]

Elementary and existential doctrines P : C op → InfSL is elementary and existential if it has “direct images”: i.e. for all f : X → A , there is a covariant ‘natural’ assignment ∃ f : P ( X ) → P ( A ) such that ∃ f ( α ) ≤ β α ≤ P ( f )( β ) Equality: � id A , id A � : A − → A × A [F.W. Lawvere]

Elementary and existential doctrines P : C op → InfSL is elementary and existential if it has “direct images”: i.e. for all f : X → A , there is a covariant ‘natural’ assignment ∃ f : P ( X ) → P ( A ) such that ∃ f ( α ) ≤ β α ≤ P ( f )( β ) � P ( A × A ) Equality: � id A , id A � : A − → A × A ∃ � id A , id A � : P ( A ) � δ A ⊤ ✤ [F.W. Lawvere]

Elementary and existential doctrines P : C op → InfSL is elementary and existential if it has “direct images”: i.e. for all f : X → A , there is a covariant ‘natural’ assignment ∃ f : P ( X ) → P ( A ) such that ∃ f ( α ) ≤ β α ≤ P ( f )( β ) � P ( A × A ) Equality: � id A , id A � : A − → A × A ∃ � id A , id A � : P ( A ) � δ A ⊤ ✤ When P is P δ A = { ( a , b ) ∈ A × A | a = b } [F.W. Lawvere]

Triposes P : C op → InfSL existential and elementary.

Triposes P : C op → InfSL existential and elementary. P �→ C [ P ]: Tripos → Topos Hyland, Johnstone, Pitts. Tripos Theory. Math. Proc. Camb. Phil. Soc. 1980.

Triposes P : C op → InfSL existential and elementary. P �→ C [ P ]: Tripos → Topos Hyland, Johnstone, Pitts. Tripos Theory. Math. Proc. Camb. Phil. Soc. 1980. P �→ C [ P ]: EED → Xct Pitts. Tripos Theory in retrospect. Math. Structures. Comput. Sci. 2002.

� � Triposes P : C op → InfSL existential and elementary. P �→ C [ P ]: Tripos → Topos Hyland, Johnstone, Pitts. Tripos Theory. Math. Proc. Camb. Phil. Soc. 1980. P �→ C [ P ]: EED → Xct Pitts. Tripos Theory in retrospect. Math. Structures. Comput. Sci. 2002. � CEED � Xct EED ⊥ ⊥ Maietti, Rosolini. Unifying exact completions. Appl. Categ. Structures, 2015.

� Triposes P : C op → InfSL existential and elementary. P �→ C [ P ]: Tripos → Topos Hyland, Johnstone, Pitts. Tripos Theory. Math. Proc. Camb. Phil. Soc. 1980. P �→ C [ P ]: EED → Xct Pitts. Tripos Theory in retrospect. Math. Structures. Comput. Sci. 2002. � Xtc EED ⊥ Maietti, Rosolini. Unifying exact completions. Appl. Categ. Structures, 2015.

Comprehension schema and effective quotients

Comprehension schema and effective quotients P : Set op → InfSL is the powerset functor.

Comprehension schema and effective quotients P : Set op → InfSL is the powerset functor. Comprehension schema: for α ∈ P ( A ) { | α | } : { a ∈ A | a ∈ α } → A

Comprehension schema and effective quotients P : Set op → InfSL is the powerset functor. Comprehension schema: for α ∈ P ( A ) { | α | } : { a ∈ A | a ∈ α } → A Effective quotients: for an equivalence relation ρ ∈ P ( A × A ) a �→ [ a ]: A → A /ρ

Comprehension schema and effective quotients P : Set op → InfSL is the powerset functor. Comprehension schema: for α ∈ P ( A ) { | α | } : { a ∈ A | a ∈ α } → A Effective quotients: for an equivalence relation ρ ∈ P ( A × A ) a �→ [ a ]: A → A /ρ Abstract characterization in the framework of doctrines.

Completions P : C op → InfSL

Completions P : C op → InfSL Comprehension completion: the comprehension schema can be freely added to any doctrine. P c : C op → InfSL c [Grothendieck’s construction of vertical morphisms.]

Completions P : C op → InfSL Comprehension completion: the comprehension schema can be freely added to any doctrine. P c : C op → InfSL c [Grothendieck’s construction of vertical morphisms.] Elementary quotient completion: effective quotients can be freely added to any elementary existential doctrine. P : Q op � P → InfSL [M.E. Maietti and G. Rosolini. Elementary quotient completion. 2013]

Back to triposes

� Back to triposes Tripos Topos

� Back to triposes P : C op → InfSL Tripos Topos

� � Back to triposes P : C op → InfSL Tripos ❴ P c : C op → InfSL c Topos

� � � Back to triposes P : C op → InfSL Tripos ❴ P c : C op → InfSL c ❴ � P c : Q op C c → InfSL Topos

� � � Back to triposes P : C op → InfSL Tripos ❴ P c : C op → InfSL c ❴ � P c : Q op C c → InfSL Topos Theorem: Q P c is a topos iff � P c satisfies the Rule of Unique Choice.

Rules of Choice

Rules of Choice Rule of Unique Choice: For every Total and Single valued relation R ∈ P ( A × B ) there is f : A → B such that R = P ( f × id B )( δ B )

Rules of Choice Rule of Unique Choice: For every Total and Single valued relation R ∈ P ( A × B ) there is f : A → B such that R = P ( f × id B )( δ B ) Rule of Choice: For every Total relation R ∈ P ( A × B ) there is f : A → B such that ∃ π A R = P ( � id A , f � )( R )

Rules of Choice Rule of Unique Choice: For every Total and Single valued relation R ∈ P ( A × B ) there is f : A → B such that R = P ( f × id B )( δ B ) Rule of Choice: For every Total relation R ∈ P ( A × B ) there is f : A → B such that ∃ π A R = P ( � id A , f � )( R ) Hilbert’s ǫ -operator: P has Hilbert’s ǫ -operator if for every R ∈ P ( A × B ) there is ǫ R : A → B such that ∃ π A R = P ( � id A , ǫ R � )( R )

Characterizations P : C op → InfSL is a tripos. Theorem: ˆ P satisfies the Rule of Unique Choice if and only if P satisfies the Rule of choice [Maietti & Rosolini. Relating quotient completions via categorical logic. 2016]

Characterizations P : C op → InfSL is a tripos. Theorem: ˆ P satisfies the Rule of Unique Choice if and only if P satisfies the Rule of choice [Maietti & Rosolini. Relating quotient completions via categorical logic. 2016] Theorem: P c satisfies the Rule of Choice if and only if P has Hilbert’s ǫ -operator [Maietti, Pasquali & Rosolini. Triposes, exact completions, and Hilbert’s ǫ -operator. 2017]

Characterizations P : C op → InfSL is a tripos. Theorem: ˆ P satisfies the Rule of Unique Choice if and only if P satisfies the Rule of choice [Maietti & Rosolini. Relating quotient completions via categorical logic. 2016] Theorem: P c satisfies the Rule of Choice if and only if P has Hilbert’s ǫ -operator [Maietti, Pasquali & Rosolini. Triposes, exact completions, and Hilbert’s ǫ -operator. 2017] Corollary: Q ˆ P c is a topos if and only if P has Hilbert’s ǫ -operator

Examples and future developments W is a poset, ⊥ ∈ W , L = W op is a well order.

� � Examples and future developments W is a poset, ⊥ ∈ W , L = W op is a well order. L : Set op ∗ − → InfSL L X X α �→ α ◦ f f L Y Y

� � Examples and future developments W is a poset, ⊥ ∈ W , L = W op is a well order. L : Set op ∗ − → InfSL L X X α �→ α ◦ f f L Y Y L has Hilbert’s ǫ -operator. Q ˆ L c is the topos of sheaves over L .

� � Examples and future developments W is a poset, ⊥ ∈ W , L = W op is a well order. L : Set op ∗ − → InfSL L X X α �→ α ◦ f f L Y Y L has Hilbert’s ǫ -operator. Q ˆ L c is the topos of sheaves over L . L is not classical, but it satisfies the weak Law of Excluded Middle.