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 Minimalist Foundation to a predicative tripos From the Minimalist Foundation to a predicative tripos
From the Minimalist Foundation to a predicative tripos The Minimalist Foundation (Maietti, Sambin 2005, Maietti 2009) consists of two levels:
From the Minimalist Foundation to a predicative tripos The Minimalist Foundation (Maietti, Sambin 2005, Maietti 2009) consists of two levels: an intensional level (to extract computational contents of math)
From the Minimalist Foundation to a predicative tripos The Minimalist Foundation (Maietti, Sambin 2005, Maietti 2009) consists of two levels: an intensional level (to extract computational contents of math) an extensional level (actual mathematics)
From the Minimalist Foundation to a predicative tripos The Minimalist Foundation (Maietti, Sambin 2005, Maietti 2009) consists of two levels: an intensional level (to extract computational contents of math) an extensional level (actual mathematics) connected via a setoid interpretation
From the Minimalist Foundation to a predicative tripos The Minimalist Foundation (Maietti, Sambin 2005, Maietti 2009) consists of 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):
� � � � � � � � � � � � � � From the Minimalist Foundation to a predicative tripos The Minimalist Foundation (Maietti, Sambin 2005, Maietti 2009) consists of 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 Set � � � Col interpretation Prop s � � Prop s � � ⇐ Prop Prop
� � � � � � � � � � � � � � From the Minimalist Foundation to a predicative tripos The Minimalist Foundation (Maietti, Sambin 2005, Maietti 2009) consists of 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 Set � � � Col interpretation Prop s � � Prop s � � ⇐ Prop Prop idea : mimic this structure to define a predicative version of the effective topos from a predicative effective tripos.
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).
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 ̂ ID 1
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 ̂ ID 1 via a variant of Martin-L¨ of type theory (props-as-types)
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 ̂ ID 1 via a variant of Martin-L¨ of type theory (props-as-types) not a categorical model: problems with interpretation of λ -abstraction and substitution (weak exponentials)
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 ̂ ID 1 via a variant of Martin-L¨ of 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
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 ̂ ID 1 via a variant of Martin-L¨ of 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
The predicative effective tripos The predicative effective tripos
The predicative effective tripos The base category C r :
The predicative effective tripos The base category C r : obj A ∶= { x ∣ ϕ A ( x )} , ϕ A ( x ) formula of ̂ ID 1 as usual x ε A is ϕ A ( x ) ;
The predicative effective tripos The base category C r : obj A ∶= { x ∣ ϕ A ( x )} , ϕ A ( x ) formula of ̂ ID 1 as usual x ε A is ϕ A ( x ) ; arr [ n ] ≈ A , B ∶ A → B
The predicative effective tripos The base category C r : obj A ∶= { x ∣ ϕ A ( x )} , ϕ A ( x ) formula of ̂ ID 1 as usual x ε A is ϕ A ( x ) ; arr [ n ] ≈ A , B ∶ A → B n numeral;
The predicative effective tripos The base category C r : obj A ∶= { x ∣ ϕ A ( x )} , ϕ A ( x ) formula of ̂ ID 1 as usual x ε A is ϕ A ( x ) ; arr [ n ] ≈ A , B ∶ A → B n numeral; x ε A ⊢ ̂ ID1 { n }( x ) ε B;
The predicative effective tripos The base category C r : obj A ∶= { x ∣ ϕ A ( x )} , ϕ A ( x ) formula of ̂ ID 1 as usual x ε A is ϕ A ( 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 ) .
The predicative effective tripos The base category C r : obj A ∶= { x ∣ ϕ A ( x )} , ϕ A ( x ) formula of ̂ ID 1 as usual x ε A is ϕ A ( 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 ) . C r is a finitely complete weakly locally cartesian closed category with parameterized list objects.
The predicative effective tripos Collections over C r Define over C r an indexed category representing dependent collections Col r ∶ C op → Cat r
The predicative effective tripos Collections over C r Define over C r an indexed category representing dependent collections Col r ∶ C op → Cat r Col r ( A ) :
The predicative effective tripos Collections over C r Define over C r an indexed category representing dependent collections Col r ∶ C op → Cat r Col r ( A ) : objects : definable classes with a parameter over the context A
The predicative effective tripos Collections over C r Define over C r an indexed category representing dependent collections Col r ∶ C op → Cat r Col r ( A ) : objects : definable classes with a parameter over the context A arrows : recursive functions (possibly depending on the context) represented by numerals
The predicative effective tripos Collections over C r Define over C r an indexed category representing dependent collections Col r ∶ C op → Cat r Col r ( A ) : objects : definable classes with a parameter over the context A arrows : recursive functions (possibly depending on the context) represented by numerals Col r ([ n ]) are substitution functors.
The predicative effective tripos Collections over C r Define over C r an indexed category representing dependent collections Col r ∶ C op → Cat r Col r ( A ) : objects : definable classes with a parameter over the context A arrows : recursive functions (possibly depending on the context) represented by numerals Col r ([ n ]) are substitution functors.
The predicative effective tripos Propositions over C r Define over C r a first-order hyperdoctrine Prop r ∶ C op → Heyt r
The predicative effective tripos Propositions over C r Define over C r a first-order hyperdoctrine Prop r ∶ C op → Heyt r the posetal reflection of the doctrine of Kleene realizability for which
The predicative effective tripos Propositions over C r Define over C r a first-order hyperdoctrine Prop r ∶ C op → Heyt r 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 ε A
The predicative effective tripos Propositions over C r Define over C r a first-order hyperdoctrine Prop r ∶ C op → Heyt r 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 ε A 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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