Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Yet another approach to semantics? Why add yet another proposal?! Natural models are slick! Straightforward interpretation of syntax; Avoids reference to (e.g.) fibrations; Also avoids heavy structure seen in (e.g.) categories with families; Flexibility—can study natural models from several viewpoints. They elucidate hidden structure: Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Yet another approach to semantics? Why add yet another proposal?! Natural models are slick! Straightforward interpretation of syntax; Avoids reference to (e.g.) fibrations; Also avoids heavy structure seen in (e.g.) categories with families; Flexibility—can study natural models from several viewpoints. They elucidate hidden structure: Connection with polynomial functors; Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Yet another approach to semantics? Why add yet another proposal?! Natural models are slick! Straightforward interpretation of syntax; Avoids reference to (e.g.) fibrations; Also avoids heavy structure seen in (e.g.) categories with families; Flexibility—can study natural models from several viewpoints. They elucidate hidden structure: Connection with polynomial functors; 1 + Σ + Π � polynomial monad; Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Yet another approach to semantics? Why add yet another proposal?! Natural models are slick! Straightforward interpretation of syntax; Avoids reference to (e.g.) fibrations; Also avoids heavy structure seen in (e.g.) categories with families; Flexibility—can study natural models from several viewpoints. They elucidate hidden structure: Connection with polynomial functors; 1 + Σ + Π � polynomial monad; Natural structure of a double category. Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Dependent type theory 1 Natural models of type theory 2 Algebraic description of homomorphisms 3 Functorial description of homomorphisms 4 5 Interpreting the syntax Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Representable natural transformations U : C op → Set . A natural Let C be a category and let U , � transformation p : � U → U is representable if, for all Γ ∈ ob ( C ) and all A ∈ U (Γ) , q Γ q Γ A A � y (Γ · A ) U p y ( p Γ y ( p Γ A ) A ) y (Γ) U A Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Representable natural transformations U : C op → Set . A natural Let C be a category and let U , � transformation p : � U → U is representable if, for all Γ ∈ ob ( C ) and all A ∈ U (Γ) , there exist Γ · A , p Γ A , q Γ A making the following diagram a pullback: q Γ q Γ A A � y (Γ · A ) U p y ( p Γ y ( p Γ A ) A ) y (Γ) U A Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Informal semantics Type theory Representable natural transformation Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Informal semantics Type theory Representable natural transformation Γ context Γ ∈ ob ( C ) Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Informal semantics Type theory Representable natural transformation Γ context Γ ∈ ob ( C ) Γ ⊢ A A ∈ U (Γ) Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Informal semantics Type theory Representable natural transformation Γ context Γ ∈ ob ( C ) Γ ⊢ A A ∈ U (Γ) � U a Γ ⊢ a : A p y (Γ) U A Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Informal semantics Type theory Representable natural transformation Γ context Γ ∈ ob ( C ) Γ ⊢ A A ∈ U (Γ) � U a Γ ⊢ a : A p y (Γ) U A ∆ ⊢ γ : Γ γ : ∆ → Γ in C Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Informal semantics Type theory Representable natural transformation Γ context Γ ∈ ob ( C ) Γ ⊢ A A ∈ U (Γ) � U a Γ ⊢ a : A p y (Γ) U A ∆ ⊢ γ : Γ γ : ∆ → Γ in C a { γ } � U ∆ ⊢ a { γ } : A { γ } p a y (∆) y (Γ) U y ( γ ) A A { γ } Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Context extension ↔ representability Type theory: Given the following: Γ ⊢ A , ∆ ⊢ γ : Γ , ∆ ⊢ a : A { γ } There is a unique substitution ∆ ⊢ � γ, a � : Γ · A , such that ∆ ⊢ p Γ Γ ⊢ q Γ A ◦ � γ, a � = γ : Γ and A {� γ, a �} = a : A Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Context extension ↔ representability Type theory: Given the following: Γ ⊢ A , ∆ ⊢ γ : Γ , ∆ ⊢ a : A { γ } There is a unique substitution ∆ ⊢ � γ, a � : Γ · A , such that ∆ ⊢ p Γ Γ ⊢ q Γ A ◦ � γ, a � = γ : Γ and A {� γ, a �} = a : A Representable natural transformation: y (∆) a y ( � γ, a � ) q Γ � A y (Γ · A ) U � y ( γ ) p y ( p Γ A ) y (Γ) U A Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Connection with categories with families Theorem (Awodey, 2015) Specifying a category with families with base category C is equivalent to specifying a representable natural transformation between presheaves on C . Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Connection with categories with families Theorem (Awodey, 2015) Specifying a category with families with base category C is equivalent to specifying a representable natural transformation between presheaves on C . A natural model is a representable natural transformation. Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Connection with categories with families Theorem (Awodey, 2015) Specifying a category with families with base category C is equivalent to specifying a representable natural transformation between presheaves on C . A natural model is a representable natural transformation. Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Connection with categories with families Theorem (Awodey, 2015) Specifying a category with families with base category C is equivalent to specifying a representable natural transformation between presheaves on C . A natural model is a representable natural transformation. We seek an essentially algebraic definition. Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Representability via categories of elements Lemma A natural transformation p : � U → U is representable if and only if the � � � C � induced functor on categories of elements C p : U → C U has a right adjoint p ∗ . Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Representability via categories of elements Lemma A natural transformation p : � U → U is representable if and only if the � � � C � induced functor on categories of elements C p : U → C U has a right adjoint p ∗ . p ∗ (Γ , A ) = (Γ · A , q Γ A ) Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Representability via categories of elements Lemma A natural transformation p : � U → U is representable if and only if the � � � C � induced functor on categories of elements C p : U → C U has a right adjoint p ∗ . p ∗ (Γ , A ) = (Γ · A , q Γ A ) ε (Γ , A ) = p Γ A : Γ · A → A Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Representability via categories of elements Lemma A natural transformation p : � U → U is representable if and only if the � � � C � induced functor on categories of elements C p : U → C U has a right adjoint p ∗ . p ∗ (Γ , A ) = (Γ · A , q Γ A ) ε (Γ , A ) = p Γ A : Γ · A → A γ : ∆ → Γ � p ∗ ( γ ) as follows: p ∗ ( γ ) ∆ · A { γ } Γ · A � p ∆ p Γ A { γ } A ∆ Γ γ Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Definition of a natural model Definition A natural model is an octuple C = ( C , ⋄ , U , � U , p , p ∗ , η, ε ) consisting of the following data: Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Definition of a natural model Definition A natural model is an octuple C = ( C , ⋄ , U , � U , p , p ∗ , η, ε ) consisting of the following data: A base category C with a terminal object ⋄ ; Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Definition of a natural model Definition A natural model is an octuple C = ( C , ⋄ , U , � U , p , p ∗ , η, ε ) consisting of the following data: A base category C with a terminal object ⋄ ; U : C op → Set ; Presheaves U , � Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Definition of a natural model Definition A natural model is an octuple C = ( C , ⋄ , U , � U , p , p ∗ , η, ε ) consisting of the following data: A base category C with a terminal object ⋄ ; U : C op → Set ; Presheaves U , � Functors � p � C � U C U p ∗ such that p commutes with the projection maps to C ; Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Definition of a natural model Definition A natural model is an octuple C = ( C , ⋄ , U , � U , p , p ∗ , η, ε ) consisting of the following data: A base category C with a terminal object ⋄ ; U : C op → Set ; Presheaves U , � Functors � p � C � U C U p ∗ such that p commutes with the projection maps to C ; Natural transformations η : id → p ∗ ◦ p ε : p ◦ p ∗ → id and forming the unit and counit, respectively, of an adjunction p ⊣ p ∗ . Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Outline We’ll follow the standard pattern for functorial semantics: Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Outline We’ll follow the standard pattern for functorial semantics: Define the notion of homomorphism of natural models; Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Outline We’ll follow the standard pattern for functorial semantics: Define the notion of homomorphism of natural models; Show that the syntax for type theory on a given signature Σ presents the free natural model T on Σ ; Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Outline We’ll follow the standard pattern for functorial semantics: Define the notion of homomorphism of natural models; Show that the syntax for type theory on a given signature Σ presents the free natural model T on Σ ; � An interpretation of Σ in a natural model C is given by a homomorphism T → C . Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Dependent type theory 1 Natural models of type theory 2 Algebraic description of homomorphisms 3 Functorial description of homomorphisms 4 5 Interpreting the syntax Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Algebraic description of homomorphisms Definition Let C = ( C , ⋄ , U , � U , p , p ∗ , η, ε ) and D = ( D , • , V , � V , q , q ∗ , σ, τ ) be natural models. A homomorphism from C to D is a triple ( F , Φ , � Φ) consisting of: Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Algebraic description of homomorphisms Definition Let C = ( C , ⋄ , U , � U , p , p ∗ , η, ε ) and D = ( D , • , V , � V , q , q ∗ , σ, τ ) be natural models. A homomorphism from C to D is a triple ( F , Φ , � Φ) consisting of: A functor F : C → D ; Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Algebraic description of homomorphisms Definition Let C = ( C , ⋄ , U , � U , p , p ∗ , η, ε ) and D = ( D , • , V , � V , q , q ∗ , σ, τ ) be natural models. A homomorphism from C to D is a triple ( F , Φ , � Φ) consisting of: A functor F : C → D ; Functors � � � � � � � Φ : U → V and Φ : U → V C D C D such that. . . Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Algebraic description of homomorphisms . . . the following diagrams commute (highlighted in red): � � � � C � Φ Φ D � U V p ∗ p ∗ q ∗ q ∗ p q � � C U D V Φ C D F Action on types respects context and substitution Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Algebraic description of homomorphisms . . . the following diagrams commute (highlighted in red): � � � � C � Φ Φ D � U V p ∗ p ∗ q ∗ q ∗ p q � � C U D V Φ C D F Action on terms respects context and substitution Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Algebraic description of homomorphisms . . . the following diagrams commute (highlighted in red): � � � � C � Φ Φ D � U V p p ∗ p ∗ q q ∗ q ∗ � � C U D V Φ C D F Action on terms respects typing Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Algebraic description of homomorphisms . . . the following diagrams commute (highlighted in red): � � � � C � Φ Φ D � U V p ∗ p ∗ q ∗ q ∗ p q � � C U D V Φ C D F Action on contexts and substitutions respects context extension Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Algebraic description of homomorphisms . . . and Φ , � Φ respect the adjunctions ( p ⊣ p ∗ , η, ε ) and ( q ⊣ q ∗ , σ, τ ) , i.e. Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Algebraic description of homomorphisms . . . and Φ , � Φ respect the adjunctions ( p ⊣ p ∗ , η, ε ) and ( q ⊣ q ∗ , σ, τ ) , i.e. p ◦ p ∗ � � C U C U ε id Φ Φ q ◦ q ∗ � � D V D V τ id Fp Γ A = p F Γ Counit. Φ · ε = τ · Φ FA : F Γ · FA → F Γ � Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Algebraic description of homomorphisms . . . and Φ , � Φ respect the adjunctions ( p ⊣ p ∗ , η, ε ) and ( q ⊣ q ∗ , σ, τ ) , i.e. p ◦ p ∗ id � � � � C � C � C U C U U U η ε q ∗ ◦ q id � � Φ Φ Φ Φ q ◦ q ∗ id � � � � D � D � D V D V V V τ σ id q ∗ ◦ q Fp Γ A = p F Γ Counit. Φ · ε = τ · Φ FA : F Γ · FA → F Γ � Unit. � Φ · η = σ · � F � id Γ , q Γ A � = � id F Γ , q F Γ Φ FA � : F Γ → F Γ · FA � Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Algebraic description of homomorphisms . . . and Φ , � Φ respect the adjunctions ( p ⊣ p ∗ , η, ε ) and ( q ⊣ q ∗ , σ, τ ) , i.e. p ◦ p ∗ id � � � � C � C � C U C U U U η ε q ∗ ◦ q id � � Φ Φ Φ Φ q ◦ q ∗ id � � � � D � D � D V D V V V τ σ id q ∗ ◦ q Fp Γ A = p F Γ Counit. Φ · ε = τ · Φ FA : F Γ · FA → F Γ � Unit. � Φ · η = σ · � F � id Γ , q Γ A � = � id F Γ , q F Γ Φ FA � : F Γ → F Γ · FA � . . . and F ( ⋄ ) = • . Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Category of natural models Theorem There is a category NM , where: The objects of NM are natural models; The morphisms of NM are homomorphisms; The identity morphism on a natural model C is ( id C , id � U , id � � U ) ; Composition is given componentwise: ( G , Ψ , � Ψ) ◦ ( F , Φ , � Φ) = ( G ◦ F , Ψ ◦ Φ , � Ψ ◦ � Φ) Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Category of natural models Theorem There is a category NM , where: The objects of NM are natural models; The morphisms of NM are homomorphisms; The identity morphism on a natural model C is ( id C , id � U , id � � U ) ; Composition is given componentwise: ( G , Ψ , � Ψ) ◦ ( F , Φ , � Φ) = ( G ◦ F , Ψ ◦ Φ , � Ψ ◦ � Φ) Since homomorphisms are defined diagramatically, this is extremely simple to prove. Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Dependent type theory 1 Natural models of type theory 2 Algebraic description of homomorphisms 3 Functorial description of homomorphisms 4 5 Interpreting the syntax Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Remark on Kan extension Any functor F : C → D between small categories induces an adjunction F ! ⊣ F ∗ between presheaf categories F ! F ! Set C op Set D op F ∗ F ∗ y ∼ y = C D F where F ∗ = − ◦ F is precomposition with F ; and F ! is left Kan extension along F . Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Remark on Kan extension Any functor F : C → D between small categories induces an adjunction F ! ⊣ F ∗ between presheaf categories F ! F ! Set C op Set D op F ∗ F ∗ y ∼ y = C D F where F ∗ = − ◦ F is precomposition with F ; and F ! is left Kan extension along F . = y ◦ F : C → Set D op . Moreover, F ! ◦ y ∼ Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Functorial presentation of homomorphisms Specifying a homomorphism ( F , Φ , � Φ) : C → D is equivalent to specifying: Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Functorial presentation of homomorphisms Specifying a homomorphism ( F , Φ , � Φ) : C → D is equivalent to specifying: A functor F : C → D ; Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Functorial presentation of homomorphisms Specifying a homomorphism ( F , Φ , � Φ) : C → D is equivalent to specifying: A functor F : C → D ; ϕ : F ! � U → � Natural transformations ϕ : F ! U → V and � V Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Functorial presentation of homomorphisms Specifying a homomorphism ( F , Φ , � Φ) : C → D is equivalent to specifying: A functor F : C → D ; ϕ : F ! � U → � Natural transformations ϕ : F ! U → V and � V such that F ( ⋄ ) = • ; Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Functorial presentation of homomorphisms Specifying a homomorphism ( F , Φ , � Φ) : C → D is equivalent to specifying: A functor F : C → D ; ϕ : F ! � U → � Natural transformations ϕ : F ! U → V and � V such that F ( ⋄ ) = • ; � F ! � ϕ � U V The diagram commutes; q F ! p F ! U V ϕ Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Functorial presentation of homomorphisms Specifying a homomorphism ( F , Φ , � Φ) : C → D is equivalent to specifying: A functor F : C → D ; ϕ : F ! � U → � Natural transformations ϕ : F ! U → V and � V such that F ( ⋄ ) = • ; � F ! � ϕ � U V The diagram commutes; q F ! p F ! U V ϕ F (Γ · A ) = F Γ · FA for all Γ ∈ ob ( C ) ; and Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Functorial presentation of homomorphisms Specifying a homomorphism ( F , Φ , � Φ) : C → D is equivalent to specifying: A functor F : C → D ; ϕ : F ! � U → � Natural transformations ϕ : F ! U → V and � V such that F ( ⋄ ) = • ; � F ! � ϕ � U V The diagram commutes; q F ! p F ! U V ϕ F (Γ · A ) = F Γ · FA for all Γ ∈ ob ( C ) ; and The comparison morphisms c Γ A : F (Γ · A ) → F Γ · FA are identities. Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Action on types and terms We obtain an action of ϕ on types and � ϕ on terms as follows. Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Action on types and terms We obtain an action of ϕ on types and � ϕ on terms as follows. Action on types. A ∈ U (Γ) � FA ∈ V ( F Γ) via Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Action on types and terms We obtain an action of ϕ on types and � ϕ on terms as follows. Action on types. A ∈ U (Γ) � FA ∈ V ( F Γ) via A y (Γ) U F ! y (Γ) Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Action on types and terms We obtain an action of ϕ on types and � ϕ on terms as follows. Action on types. A ∈ U (Γ) � FA ∈ V ( F Γ) via F ! A F ! A A ϕ y (Γ) U F ! y (Γ) F ! U V ∼ ∼ � = = FA FA F ! y (Γ) y (Γ) Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Action on types and terms We obtain an action of ϕ on types and � ϕ on terms as follows. Action on types. A ∈ U (Γ) � FA ∈ V ( F Γ) via F ! A F ! A A ϕ y (Γ) U F ! y (Γ) F ! U V ∼ ∼ � = = FA FA F ! y (Γ) y (Γ) Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Action on types and terms We obtain an action of ϕ on types and � ϕ on terms as follows. Action on types. A ∈ U (Γ) � FA ∈ V ( F Γ) via F ! A F ! A A ϕ y (Γ) U F ! y (Γ) F ! U V ∼ ∼ � = = FA FA F ! y (Γ) y ( F Γ) Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Action on types and terms We obtain an action of ϕ on types and � ϕ on terms as follows. Action on types. A ∈ U (Γ) � FA ∈ V ( F Γ) via F ! A F ! A A ϕ y (Γ) U F ! y (Γ) F ! U V ∼ ∼ � = = FA FA F ! y (Γ) y ( F Γ) Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Action on types and terms We obtain an action of ϕ on types and � ϕ on terms as follows. Action on types. A ∈ U (Γ) � FA ∈ V ( F Γ) via F ! A F ! A A ϕ y (Γ) U F ! y (Γ) F ! U V ∼ ∼ � = = FA FA F ! y (Γ) y ( F Γ) Action on terms. a ∈ � U (Γ) � Fa ∈ � V ( F Γ) via F ! a � a � F ! � ϕ � y (Γ) U F ! y (Γ) U V � ∼ = Fa F ! y (Γ) y ( F Γ) Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Where the comparison morphisms c Γ A come from q F Γ y ( F Γ · FA ) FA y ( c Γ y ( c Γ A ) A ) Fq Γ F ! q Γ q Γ q Γ ∼ � = � ϕ � A A A A y ( F (Γ · A )) y (Γ · A ) U V y ( p F Γ FA ) p q F ! y ( p Γ y ( p Γ y ( p Γ F ! ( p ) A ) A ) A ) y ( Fp Γ A ) y (Γ) U V ∼ = ϕ FA FA F ! A A Set-up: A type in context Γ Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Where the comparison morphisms c Γ A come from q F Γ y ( F Γ · FA ) FA y ( c Γ y ( c Γ A ) A ) Fq Γ F ! q Γ q Γ q Γ ∼ � = � ϕ � A A A A y ( F (Γ · A )) y (Γ · A ) U V � y ( p F Γ FA ) p q F ! y ( p Γ y ( p Γ y ( p Γ F ! ( p ) A ) A ) A ) y ( Fp Γ A ) y (Γ) U V ∼ = ϕ FA FA F ! A A Context extension of Γ by A Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Where the comparison morphisms c Γ A come from q F Γ y ( F Γ · FA ) FA y ( c Γ y ( c Γ A ) A ) Fq Γ F ! q Γ q Γ q Γ ∼ � = ϕ F ! � � A A A A y ( F (Γ · A )) F ! y (Γ · A ) U V y ( p F Γ FA ) F ! y ( p Γ y ( p Γ y ( p Γ q p F ! ( p ) A ) A ) A ) y ( Fp Γ A ) F ! y (Γ) F ! U V ∼ = FA FA F ! A A ϕ Apply F ! Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Where the comparison morphisms c Γ A come from q F Γ y ( F Γ · FA ) FA y ( c Γ y ( c Γ A ) A ) Fq Γ F ! q Γ q Γ q Γ ∼ � = ϕ A A A A F ! � � y ( F (Γ · A )) F ! y (Γ · A ) U V y ( p F Γ FA ) y ( Fp Γ F ! y ( p Γ y ( p Γ y ( p Γ p q F ! ( p ) A ) A ) A ) A ) y ( F Γ) F ! y (Γ) F ! U V ∼ = FA FA F ! A A ϕ F ! ◦ y ∼ = y ( F − ) Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Where the comparison morphisms c Γ A come from q F Γ y ( F Γ · FA ) FA y ( c Γ y ( c Γ A ) A ) Fq Γ F ! q Γ q Γ q Γ ∼ � = ϕ F ! � � A A A A y ( F (Γ · A )) F ! y (Γ · A ) U V y ( p F Γ FA ) y ( Fp Γ F ! y ( p Γ y ( p Γ y ( p Γ q p F ! ( p ) A ) A ) A ) A ) y ( F Γ) F ! y (Γ) F ! U V ∼ = FA FA F ! A A ϕ Paste square for ϕ, � ϕ Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Where the comparison morphisms c Γ A come from q F Γ y ( F Γ · FA ) FA y ( c Γ y ( c Γ A ) A ) Fq Γ F ! q Γ q Γ q Γ ∼ � = ϕ A A A A F ! � � y ( F (Γ · A )) F ! y (Γ · A ) U V y ( p F Γ FA ) y ( Fp Γ q A ) F ! y ( p Γ y ( p Γ y ( p Γ p F ! ( p ) A ) A ) A ) y ( F Γ) V ∼ = FA FA A F ! A ϕ Action of ϕ on types and � ϕ on terms Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Where the comparison morphisms c Γ A come from q F Γ y ( F Γ · FA ) FA � y ( c Γ y ( c Γ A ) A ) Fq Γ F ! q Γ q Γ q Γ ∼ � = ϕ F ! � � A A A A y ( F (Γ · A )) F ! y (Γ · A ) U V y ( p F Γ FA ) y ( Fp Γ q A ) F ! y ( p Γ y ( p Γ y ( p Γ p F ! ( p ) A ) A ) A ) y ( F Γ) V ∼ = FA FA A F ! A ϕ Extend context F Γ by FA Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Where the comparison morphisms c Γ A come from q F Γ y ( F Γ · FA ) FA y ( c Γ y ( c Γ A ) A ) Fq Γ F ! q Γ q Γ q Γ ∼ � = F ! � ϕ � A A A A y ( F (Γ · A )) F ! y (Γ · A ) U V y ( p F Γ FA ) q y ( Fp Γ A ) F ! y ( p Γ y ( p Γ y ( p Γ p F ! ( p ) A ) A ) A ) y ( F Γ) V ∼ = ϕ FA FA A F ! A Obtain c Γ A : F (Γ · A ) → F Γ · FA as shown Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Where the comparison morphisms c Γ A come from q F Γ y ( F Γ · FA ) FA y ( c Γ y ( c Γ A ) A ) Fq Γ F ! q Γ q Γ q Γ ∼ � = ϕ F ! � � A A A A y ( F (Γ · A )) F ! y (Γ · A ) U V y ( p F Γ FA ) y ( Fp Γ q A ) F ! y ( p Γ y ( p Γ y ( p Γ p F ! ( p ) A ) A ) A ) y ( F Γ) V ∼ = FA FA A F ! A ϕ c Γ Fp Γ A = p F Γ Fq Γ A = q F Γ A = id ⇒ and FA . FA Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Dependent type theory 1 Natural models of type theory 2 Algebraic description of homomorphisms 3 Functorial description of homomorphisms 4 5 Interpreting the syntax Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Interpreting the syntax We take a similar approach to that of S. Castellan, P . Clairambault, P . Dybjer (2015). The idea is as follows: Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Interpreting the syntax We take a similar approach to that of S. Castellan, P . Clairambault, P . Dybjer (2015). The idea is as follows: Work in a system of type theory with four kinds of judgements Γ = Γ ′ ⊢ , ∆ ⊢ γ = γ ′ : Γ , Γ ⊢ a = a ′ : A Γ ⊢ A = A ′ , (We write Γ ⊢ instead of Γ = Γ ⊢ , and so on.) Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Interpreting the syntax We take a similar approach to that of S. Castellan, P . Clairambault, P . Dybjer (2015). The idea is as follows: Work in a system of type theory with four kinds of judgements Γ = Γ ′ ⊢ , ∆ ⊢ γ = γ ′ : Γ , Γ ⊢ a = a ′ : A Γ ⊢ A = A ′ , (We write Γ ⊢ instead of Γ = Γ ⊢ , and so on.) From the syntax, build a natural model T = ( T , [] , Ty , Tm , ty , ext , sub , proj ) called the term model of the system. Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Interpreting the syntax We take a similar approach to that of S. Castellan, P . Clairambault, P . Dybjer (2015). The idea is as follows: Work in a system of type theory with four kinds of judgements Γ = Γ ′ ⊢ , ∆ ⊢ γ = γ ′ : Γ , Γ ⊢ a = a ′ : A Γ ⊢ A = A ′ , (We write Γ ⊢ instead of Γ = Γ ⊢ , and so on.) From the syntax, build a natural model T = ( T , [] , Ty , Tm , ty , ext , sub , proj ) called the term model of the system. T will (in a suitable sense) be the free natural model supporting the derivation rules for this system. Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Example 1: basic syntax With no rules for type formation, the term model is very simple: Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks Example 1: basic syntax With no rules for type formation, the term model is very simple: T has the empty context [] as its only object and the identity substitution [] ⊢ id : [] as its only morphism; Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016
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