Natural models Polynomials Semantics End Polynomials and polynomial functors Fix a locally cartesian closed category E . f : B → A P f : E → E � X �→ � X B a a : A Call P f a polynomial endofunctor and f a polynomial . Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Polynomials and polynomial functors Fix a locally cartesian closed category E . f : B → A P f : E → E � X �→ � X B a a : A Call P f a polynomial endofunctor and f a polynomial . Officially, P f is the composite ∆ B → 1 Π f Σ A → 1 E E / B E / A E where ∆ f is pullback along f and Σ f ⊣ ∆ f ⊣ Π f . Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Cartesian morphisms of polynomials ϕ : P f ⇒ P g cartesian natural � transformation Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Cartesian morphisms of polynomials ϕ 1 B D ϕ : P f ⇒ P g f � cartesian natural g � transformation A C ϕ 0 Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Cartesian morphisms of polynomials ϕ 1 B D ϕ : P f ⇒ P g f � cartesian natural g � transformation A C ϕ 0 Theorem (Gambino & Kock) Polynomials and cartesian morphisms are the 1- and 2-cells of a bicategory ; Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Cartesian morphisms of polynomials ϕ 1 B D ϕ : P f ⇒ P g f � cartesian natural g � transformation A C ϕ 0 Theorem (Gambino & Kock) Polynomials and cartesian morphisms are the 1- and 2-cells of a bicategory ; Polynomial functors and cartesian natural transformations are the 1- and 2-cells of a 2-category ; Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Cartesian morphisms of polynomials ϕ 1 B D ϕ : P f ⇒ P g f � cartesian natural g � transformation A C ϕ 0 Theorem (Gambino & Kock) Polynomials and cartesian morphisms are the 1- and 2-cells of a bicategory ; Polynomial functors and cartesian natural transformations are the 1- and 2-cells of a 2-category ; These are biequivalent. Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Admitting a unit type Theorem (Awodey) A natural model admits a unit type ⇔ there exist � 1 , � ⋆ as in: � � • ⋆ ⋆ y ( ⋄ ) U � p y ( ⋄ ) U � � 1 1 Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Admitting a unit type Theorem (Awodey) A natural model admits a unit type ⇔ there exist � 1 , � ⋆ as in: � � • ⋆ ⋆ y ( ⋄ ) U � p y ( ⋄ ) U � � 1 1 Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Admitting a unit type Theorem (Awodey) A natural model admits a unit type ⇔ there exist � 1 , � ⋆ as in: � � • ⋆ ⋆ y ( ⋄ ) U � p y ( ⋄ ) U � � 1 1 Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Admitting a unit type Theorem (Awodey) A natural model admits a unit type ⇔ there exist � 1 , � ⋆ as in: � � • ⋆ ⋆ y ( ⋄ ) U � p y ( ⋄ ) U � � 1 1 Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Admitting a unit type Theorem (Awodey) A natural model admits a unit type ⇔ there exist � 1 , � ⋆ as in: � � • ⋆ ⋆ y ( ⋄ ) U � p y ( ⋄ ) U � � 1 1 Corollary interpretations cartesian morphisms of � of unit types polynomials 1 ⇒ p Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Admitting dependent sum types Theorem (Awodey) A natural model admits a Σ -types ⇔ there exist � Σ , � pair as in: � � � � � pair pair • [ B ( a )] U A : U B : U [ A ] a :[ A ] � p π � A : U U [ A ] U � � Σ Σ Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Admitting dependent sum types Theorem (Awodey) A natural model admits a Σ -types ⇔ there exist � Σ , � pair as in: � � � � � pair pair • [ B ( a )] U A : U B : U [ A ] a :[ A ] � p π � U [ A ] U � � Σ Σ A : U Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Admitting dependent sum types Theorem (Awodey) A natural model admits a Σ -types ⇔ there exist � Σ , � pair as in: � � � � � pair pair • [ B ( a )] U A : U B : U [ A ] a :[ A ] � p π � U [ A ] U � � Σ Σ A : U Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Admitting dependent sum types Theorem (Awodey) A natural model admits a Σ -types ⇔ there exist � Σ , � pair as in: � � � � � pair pair • [ B ( a )] U A : U B : U [ A ] a :[ A ] � p π � U [ A ] U � � Σ Σ A : U Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Admitting dependent sum types Theorem (Awodey) A natural model admits a Σ -types ⇔ there exist � Σ , � pair as in: � � � � � pair pair • [ B ( a )] U A : U B : U [ A ] a :[ A ] � p π � U [ A ] U � � Σ Σ A : U Note also that P π = P p ◦ P p . Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Admitting dependent sum types Theorem (Awodey) A natural model admits a Σ -types ⇔ there exist � Σ , � pair as in: � � � � � pair pair • [ B ( a )] U A : U B : U [ A ] a :[ A ] � p π � U [ A ] U � � Σ Σ A : U Note also that P π = P p ◦ P p . Corollary interpretations cartesian morphisms of � of Σ -types polynomials p · p ⇒ p Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Admitting dependent product types Theorem (Awodey) A natural model admits a Π -types ⇔ there exist � Π , � λ as in: � [ A ] � � • • λ λ U U A : U � � � p p [ A ] p [ A ] A : U A : U � A : U U [ A ] U � � Π Π Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Admitting dependent product types Theorem (Awodey) A natural model admits a Π -types ⇔ there exist � Π , � λ as in: � [ A ] � � • • λ λ U U A : U � � � p p [ A ] p [ A ] A : U A : U � U [ A ] U � � Π Π A : U Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Admitting dependent product types Theorem (Awodey) A natural model admits a Π -types ⇔ there exist � Π , � λ as in: � [ A ] � � • • λ λ U U A : U � � � p p [ A ] p [ A ] A : U A : U � U [ A ] U � � Π Π A : U Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Admitting dependent product types Theorem (Awodey) A natural model admits a Π -types ⇔ there exist � Π , � λ as in: � [ A ] � � • • λ λ U U A : U � � � p p [ A ] p [ A ] A : U A : U � U [ A ] U � � Π Π A : U Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Admitting dependent product types Theorem (Awodey) A natural model admits a Π -types ⇔ there exist � Π , � λ as in: � [ A ] � � • • λ λ U U A : U � � � p p [ A ] p [ A ] A : U A : U � U [ A ] U � � Π Π A : U Corollary interpretations cartesian morphisms of � of Π -types polynomials P p ( p ) ⇒ p Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Monad and algebra? In summary: n.m. admits. . . ⇔ ∃ cartesian . . . Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Monad and algebra? In summary: n.m. admits. . . ⇔ ∃ cartesian . . . 1 1 ⇒ p Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Monad and algebra? In summary: n.m. admits. . . ⇔ ∃ cartesian . . . 1 1 ⇒ p Σ p · p ⇒ p Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Monad and algebra? In summary: n.m. admits. . . ⇔ ∃ cartesian . . . 1 1 ⇒ p Σ p · p ⇒ p Π P ( p ) ⇒ p Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Monad and algebra? In summary: n.m. admits. . . ⇔ ∃ cartesian . . . 1 1 ⇒ p Σ p · p ⇒ p Π P ( p ) ⇒ p This is a monad and an algebra. Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Monad and algebra? In summary: n.m. admits. . . ⇔ ∃ cartesian . . . 1 1 ⇒ p Σ p · p ⇒ p Π P ( p ) ⇒ p This is almost a monad and an algebra. Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Monad and algebra? In summary: n.m. admits. . . ⇔ ∃ cartesian . . . 1 1 ⇒ p Σ p · p ⇒ p Π P ( p ) ⇒ p This is almost a monad and an algebra. Goal. Find the appropriate notion of 3-cell (morphism of morphisms of polynomials) allowing us to make this more precise. Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Full internal subcategories Given any morphism f : B → A in a locally cartesian closed category E , we can form the full internal subcategory S ( f ) ∈ Cat ( E ) . Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Full internal subcategories Given any morphism f : B → A in a locally cartesian closed category E , we can form the full internal subcategory S ( f ) ∈ Cat ( E ) . Object of objects = A ; Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Full internal subcategories Given any morphism f : B → A in a locally cartesian closed category E , we can form the full internal subcategory S ( f ) ∈ Cat ( E ) . Object of objects = A ; � B B a Object of morphisms = a ′ a , a ′ ∈ A Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Full internal subcategories Given any morphism f : B → A in a locally cartesian closed category E , we can form the full internal subcategory S ( f ) ∈ Cat ( E ) . Object of objects = A ; � 1 f over A × A . B B a 2 f ) π ∗ a ′ = ( π ∗ Object of morphisms = a , a ′ ∈ A Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Full internal subcategories Given any morphism f : B → A in a locally cartesian closed category E , we can form the full internal subcategory S ( f ) ∈ Cat ( E ) . Object of objects = A ; � 1 f over A × A . B B a 2 f ) π ∗ a ′ = ( π ∗ Object of morphisms = a , a ′ ∈ A Cartesian morphisms of polynomials ϕ : f ⇒ g induce full and faithful functors S ( ϕ ) : S ( f ) → S ( g ) . Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Full internal subcategories Given any morphism f : B → A in a locally cartesian closed category E , we can form the full internal subcategory S ( f ) ∈ Cat ( E ) . Object of objects = A ; � 1 f over A × A . B B a 2 f ) π ∗ a ′ = ( π ∗ Object of morphisms = a , a ′ ∈ A Cartesian morphisms of polynomials ϕ : f ⇒ g induce full and faithful functors S ( ϕ ) : S ( f ) → S ( g ) . Idea: Given cartesian morphisms ϕ, ψ : f ⇒ g , take internal natural transformations S ( ϕ ) ⇒ S ( ψ ) to be our 3-cells. Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Full internal subcategories Theorem With respect to this notion of 3 -cell: Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Full internal subcategories Theorem With respect to this notion of 3 -cell: p admits 1 , Σ ⇐ ⇒ p is a pseudomonad Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Full internal subcategories Theorem With respect to this notion of 3 -cell: p admits 1 , Σ ⇐ ⇒ p is a pseudomonad p also admits Π ⇐ ⇒ p is a p- pseudoalgebra Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Full internal subcategories Theorem With respect to this notion of 3 -cell: p admits 1 , Σ ⇐ ⇒ p is a pseudomonad p also admits Π ⇐ ⇒ p is a p- pseudoalgebra • U → U , let U = S ( p ) ∈ Cat ( � Aside: For a natural model p : C ) . Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Full internal subcategories Theorem With respect to this notion of 3 -cell: p admits 1 , Σ ⇐ ⇒ p is a pseudomonad p also admits Π ⇐ ⇒ p is a p- pseudoalgebra • U → U , let U = S ( p ) ∈ Cat ( � Aside: For a natural model p : C ) . Object of objects = U . Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Full internal subcategories Theorem With respect to this notion of 3 -cell: p admits 1 , Σ ⇐ ⇒ p is a pseudomonad p also admits Π ⇐ ⇒ p is a p- pseudoalgebra • U → U , let U = S ( p ) ∈ Cat ( � Aside: For a natural model p : C ) . Object of objects = U . Object of morphisms = � A , B : U [ B ] [ A ] . Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Full internal subcategories Theorem With respect to this notion of 3 -cell: p admits 1 , Σ ⇐ ⇒ p is a pseudomonad p also admits Π ⇐ ⇒ p is a p- pseudoalgebra • U → U , let U = S ( p ) ∈ Cat ( � Aside: For a natural model p : C ) . Object of objects = U . Object of morphisms = � A , B : U [ B ] [ A ] . Considered as an indexed category C op → Cat , U is equivalent to the ‘context-indexed category of types’ of Clairambault & Dybjer (2011). Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Full internal subcategories Theorem With respect to this notion of 3 -cell: p admits 1 , Σ ⇐ ⇒ p is a pseudomonad p also admits Π ⇐ ⇒ p is a p- pseudoalgebra • U → U , let U = S ( p ) ∈ Cat ( � Aside: For a natural model p : C ) . Object of objects = U . Object of morphisms = � A , B : U [ B ] [ A ] . Considered as an indexed category C op → Cat , U is equivalent to the ‘context-indexed category of types’ of Clairambault & Dybjer (2011). Bonus: If ( C , p ) admits 1 , Σ , Π , then U is cartesian closed. Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Natural models 1 2 Connection with polynomial functors 3 Natural model semantics 4 Concluding remarks Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Initiality of the syntax Idea (Initiality ‘conjecture’) The syntax of a dependent type theory T should itself have the structure of a natural model, which is initial amongst all natural models interpreting T . Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Initiality of the syntax Idea (Initiality ‘conjecture’) The syntax of a dependent type theory T should itself have the structure of a natural model, which is initial amongst all natural models interpreting T . Goals: Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Initiality of the syntax Idea (Initiality ‘conjecture’) The syntax of a dependent type theory T should itself have the structure of a natural model, which is initial amongst all natural models interpreting T . Goals: Build the syntactic natural models for some basic type theories and prove that they satisfy an appropriate universal property; Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Initiality of the syntax Idea (Initiality ‘conjecture’) The syntax of a dependent type theory T should itself have the structure of a natural model, which is initial amongst all natural models interpreting T . Goals: Build the syntactic natural models for some basic type theories and prove that they satisfy an appropriate universal property; Expand to more complicated type theories by (algebraically) freely adding type theoretic structure. Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #1: set of basic types We’ll construct the free natural model on the theory with an I -indexed family of basic types. Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #1: set of basic types We’ll construct the free natural model on the theory with an I -indexed family of basic types. Definition • Define ( C I , p I : U I → U I ) as follows: Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #1: set of basic types We’ll construct the free natural model on the theory with an I -indexed family of basic types. Definition • Define ( C I , p I : U I → U I ) as follows: Category of contexts: C I = ( Fin / I ) op Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #1: set of basic types We’ll construct the free natural model on the theory with an I -indexed family of basic types. Definition • Define ( C I , p I : U I → U I ) as follows: Category of contexts: C I = ( Fin / I ) op u Presheaf of types: U I = cod : Fin / I → Set ( A → I ) �→ I − Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #1: set of basic types We’ll construct the free natural model on the theory with an I -indexed family of basic types. Definition • Define ( C I , p I : U I → U I ) as follows: Category of contexts: C I = ( Fin / I ) op u Presheaf of types: U I = cod : Fin / I → Set ( A → I ) �→ I − • u Presheaf of terms: U I = dom : Fin / I → Set ( A → I ) �→ A − Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #1: set of basic types We’ll construct the free natural model on the theory with an I -indexed family of basic types. Definition • Define ( C I , p I : U I → U I ) as follows: Category of contexts: C I = ( Fin / I ) op u Presheaf of types: U I = cod : Fin / I → Set ( A → I ) �→ I − • u Presheaf of terms: U I = dom : Fin / I → Set ( A → I ) �→ A − Typing map: ( p I ) A → I = u : A → I u − Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #1: set of basic types We’ll construct the free natural model on the theory with an I -indexed family of basic types. Definition • Define ( C I , p I : U I → U I ) as follows: Category of contexts: C I = ( Fin / I ) op u Presheaf of types: U I = cod : Fin / I → Set ( A → I ) �→ I − • u Presheaf of terms: U I = dom : Fin / I → Set ( A → I ) �→ A − Typing map: ( p I ) A → I = u : A → I u − u Representability data: given A − → I and i ∈ U I ( u ) = I , let u [ u , i ] ( A − → I ) • i = ( A + 1 − − → I ) Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #1: set of basic types We’ll construct the free natural model on the theory with an I -indexed family of basic types. Definition • Define ( C I , p I : U I → U I ) as follows: Category of contexts: C I = ( Fin / I ) op u Presheaf of types: U I = cod : Fin / I → Set ( A → I ) �→ I − • u Presheaf of terms: U I = dom : Fin / I → Set ( A → I ) �→ A − Typing map: ( p I ) A → I = u : A → I u − u Representability data: given A − → I and i ∈ U I ( u ) = I , let u [ u , i ] ( A − → I ) • i = ( A + 1 − − → I ) p i : A ֒ → A + 1 Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #1: set of basic types We’ll construct the free natural model on the theory with an I -indexed family of basic types. Definition • Define ( C I , p I : U I → U I ) as follows: Category of contexts: C I = ( Fin / I ) op u Presheaf of types: U I = cod : Fin / I → Set ( A → I ) �→ I − • u Presheaf of terms: U I = dom : Fin / I → Set ( A → I ) �→ A − Typing map: ( p I ) A → I = u : A → I u − u Representability data: given A − → I and i ∈ U I ( u ) = I , let u [ u , i ] ( A − → I ) • i = ( A + 1 − − → I ) p i : A ֒ → A + 1 q i = ⋆ Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #1: set of basic types Theorem ( C I , p I ) is a natural model Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #1: set of basic types Theorem • ( C I , p I ) is a natural model, and for all natural models ( C , p : U → U ) and all I-indexed families { O i } i ∈ I ⊆ U ( ⋄ ) , Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #1: set of basic types Theorem • ( C I , p I ) is a natural model, and for all natural models ( C , p : U → U ) and all I-indexed families { O i } i ∈ I ⊆ U ( ⋄ ) , there is a unique F : ( C I , p I ) → ( C , p ) with F ( i ) = O i for all i ∈ I. Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #2: freely admitting Σ -types Goal: Given a natural model ( C , p ) , construct the ‘smallest’ natural model ( C Σ , p Σ ) which extends ( C , p ) and admits Σ -types. Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #2: freely admitting Σ -types We can represent (iterated) Σ -types by binary trees. Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #2: freely admitting Σ -types We can represent (iterated) Σ -types by binary trees. � � E ( x , y , z , w ) � �� x , y � , z � : C ( x , y ) w : D ( x , y , z ) � x , y � : � B ( x ) x : A Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #2: freely admitting Σ -types We can represent (iterated) Σ -types by binary trees. � � E ( x , y , z , w ) � �� x , y � , z � : C ( x , y ) w : D ( x , y , z ) � x , y � : � B ( x ) x : A • • • • C D E A B Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #2: freely admitting Σ -types We can represent (iterated) Σ -types by binary trees. � � E ( x , y , z , w ) � �� x , y � , z � : C ( x , y ) w : D ( x , y , z ) � x , y � : � B ( x ) x : A • • • • C D E A B Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #2: freely admitting Σ -types We can represent (iterated) Σ -types by binary trees. � � E ( x , y , z , w ) � �� x , y � , z � : C ( x , y ) w : D ( x , y , z ) � x , y � : � B ( x ) x : A • • • • C D E A B Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #2: freely admitting Σ -types We can represent (iterated) Σ -types by binary trees. � � E ( x , y , z , w ) � �� x , y � , z � : C ( x , y ) w : D ( x , y , z ) � x , y � : � B ( x ) x : A • • • • C D E A B Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #2: freely admitting Σ -types We can represent (iterated) Σ -types by binary trees. � � E ( x , y , z , w ) � �� x , y � , z � : C ( x , y ) w : D ( x , y , z ) � x , y � : � B ( x ) x : A • • • • C D E A B Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #2: freely admitting Σ -types Definition • Define ( C Σ , p Σ : U Σ → U Σ ) as follows: Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #2: freely admitting Σ -types Definition • Define ( C Σ , p Σ : U Σ → U Σ ) as follows: C Σ : Objects (contexts) are the objects of C ‘formally extended’ by trees of dependent types; Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #2: freely admitting Σ -types Definition • Define ( C Σ , p Σ : U Σ → U Σ ) as follows: C Σ : Objects (contexts) are the objects of C ‘formally extended’ by trees of dependent types; U Σ is the presheaf of type trees ; Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #2: freely admitting Σ -types Definition • Define ( C Σ , p Σ : U Σ → U Σ ) as follows: C Σ : Objects (contexts) are the objects of C ‘formally extended’ by trees of dependent types; U Σ is the presheaf of type trees ; • U Σ is the presheaf of term trees ; Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
Natural models Polynomials Semantics End Example #2: freely admitting Σ -types Definition • Define ( C Σ , p Σ : U Σ → U Σ ) as follows: C Σ : Objects (contexts) are the objects of C ‘formally extended’ by trees of dependent types; U Σ is the presheaf of type trees ; • U Σ is the presheaf of term trees ; p Σ : ( tree of terms ) �→ ( tree of their types ) . Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq
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