Natural Model Semantics of Comonadic Modal Type Theory Colin Zwanziger Department of Philosophy Carnegie Mellon University August 15, 2019 at the International Conference on Homotopy Type Theory Carnegie Mellon University Zwanziger (CMU) Natural Models ˆ Comonads 1 / 31
Introduction Problem: Semantics of Comonadic Type Theory Comonads are pervasive. So comonadic dependent type theory (NPP 2008, Shulman 2018) has many intended models, e.g. : ‚ (Higher) Grothendieck toposes ` ∆Γ (Shulman 2018, 2019) ‚ In particular, cubical sets ` the 0-skeleton (LOPS 2018) ‚ Groupoids ` discretization ( cf. Zwanziger 2018) What about a general categorical semantics for comonadic DTT? Zwanziger (CMU) Natural Models ˆ Comonads 2 / 31
Introduction A Solution: Morphisms of Natural Models ‚ Simple picture: the comonadic operator is interpreted as a morphism of models of DTT that is a comonad ‚ I will work with morphisms of natural models. Zwanziger (CMU) Natural Models ˆ Comonads 3 / 31
Introduction A Solution: Morphisms of Natural Models ‚ Simple picture: the comonadic operator is interpreted as a morphism of models of DTT that is a comonad ‚ I will work with morphisms of natural models. ‚ Natural models are a nice categorical characterization of categories with families (CwFs) (Awodey 2012, 2018, Fiore 2012) ‚ The relevant morphisms of natural models and CwFs were developed by Newstead (2018) and BCMMPS (2018), respectively. Zwanziger (CMU) Natural Models ˆ Comonads 3 / 31
Introduction Morphism Semantics to Date ‚ BCMMPS use morphisms of CwFs to interpret DTT with an endo-adjunction. ‚ In Zwanziger (2019): morphisms of natural models for DTT with an adjunction. ‚ Same approach for comonadic DTT presently. So morphisms of NMs/CwFs have a broader applicability than the comonadic case. Zwanziger (CMU) Natural Models ˆ Comonads 4 / 31
Introduction Outline Introduction 1 Natural Model Theory 2 Objects Morphisms Comonadic Type Theory 3 Semantics of Comonadic Type Theory 4 Cartesian Comonads on Natural Models Interpretation Zwanziger (CMU) Natural Models ˆ Comonads 5 / 31
Natural Model Theory Objects Outline Introduction 1 Natural Model Theory 2 Objects Morphisms Comonadic Type Theory 3 Semantics of Comonadic Type Theory 4 Cartesian Comonads on Natural Models Interpretation Zwanziger (CMU) Natural Models ˆ Comonads 6 / 31
Natural Model Theory Objects Natural Models Definition (Awodey, Fiore 2012) A natural model consists of a category C a distinguished terminal object 1 P C presheaves Ty , Tm : C op Ñ Set a representable natural transformation p : Tm Ñ Ty Zwanziger (CMU) Natural Models ˆ Comonads 7 / 31
Natural Model Theory Objects Conventions Convention An object Γ P C is a “context”. An element A P Ty p Γ q is a “type in context Γ ”. An element a P Tm p Γ q such that p Γ p a q “ A is a “term a of type A in context Γ ”. Zwanziger (CMU) Natural Models ˆ Comonads 8 / 31
Natural Model Theory Objects Conventions Convention An object Γ P C is a “context”. An element A P Ty p Γ q is a “type in context Γ ”. An element a P Tm p Γ q such that p Γ p a q “ A is a “term a of type A in context Γ ”. This last is represented by the following commutative diagram: Tm a p y Γ Ty A Below, as here, we will freely use the Yoneda lemma to identify presheaf elements x P P p C q with the corresponding map x : y C Ñ P . Zwanziger (CMU) Natural Models ˆ Comonads 8 / 31
Natural Model Theory Objects Comprehension as Representability Representability of p : Tm Ñ Ty means the following: Definition Given a context Γ P C and a type A P Ty p Γ q in the context Γ , there is Γ . A P C, p A : Γ . A Ñ Γ , and v A : y p Γ . A q Ñ Tm such that the following diagram is a pullback: v A y p Γ . A q Tm { p y p A y Γ Ty A These Γ . A , p A , v A constitute the comprehension of A. Zwanziger (CMU) Natural Models ˆ Comonads 9 / 31
Natural Model Theory Objects Terms vs. Sections Remark Terms are interchangeable with a “comprehension” as sections, as depicted by the following: v A y p Γ . A q Tm Tm { a ð ñ p p y p A a y Γ Ty y Γ Ty A A See Awodey (2018) for more on natural models. Zwanziger (CMU) Natural Models ˆ Comonads 10 / 31
Natural Model Theory Morphisms Introduction 1 Natural Model Theory 2 Objects Morphisms Comonadic Type Theory 3 Semantics of Comonadic Type Theory 4 Cartesian Comonads on Natural Models Interpretation Zwanziger (CMU) Natural Models ˆ Comonads 11 / 31
Natural Model Theory Morphisms Lax Morphisms Definition A lax morphism of natural models F : C Ñ D consists of: a functor, also denoted F : C Ñ D, between the underlying categories a natural transformation φ Ty : F ! Ty C Ñ Ty D a natural transformation φ Tm : F ! Tm C Ñ Tm D such that the following diagram commutes: φ Tm F ! Tm C Tm D p D F ! p C F ! Ty C Ty D φ Ty The definitions of this section are essentially those of Newstead (2018). Zwanziger (CMU) Natural Models ˆ Comonads 12 / 31
Natural Model Theory Morphisms Notation Convention Given a lax morphism F : C Ñ D, and a type A P Ty p Γ q in context Γ P C, we write F { A for the composite φ Ty F ! A y F Γ – F ! y Γ F ! Ty C Ty D Similarly, given a term a P Tm p Γ q , we write F { a for the composite F ! a φ Tm y F Γ – F ! y Γ F ! Tm C Tm D One may think of F { A and F { a as the results of applying the morphism F to A and a . These operations are implicated in the interpretation of (respectively) formation and introduction rules for modal type operators. Zwanziger (CMU) Natural Models ˆ Comonads 13 / 31
Natural Model Theory Morphisms Lax Preservation of Context Extension Remark Let F : C Ñ D be a lax morphism. Then, given a type A P Ty C p Γ q in context Γ P C, there is a unique comparison map τ A : F p Γ . A q Ñ F Γ . p F { A q such that Fp A “ p F { A ˝ τ A and F { v A “ v F { A ˝ y p τ A q , i.e., such that the following diagram commutes: F { v A y p F p Γ . A qq y p τ A q v F { A y p F Γ . p F { A qq Tm D { y p Fp A q y p p F { A q p D y p F Γ q Ty D F { A Zwanziger (CMU) Natural Models ˆ Comonads 14 / 31
Natural Model Theory Morphisms Morphisms Definition Let F : C Ñ D be a lax morphism. Then F is said to preserve context extension if, for each type A P Ty C p Γ q in each context Γ P C, the comparison map τ A : F p Γ . A q Ñ F p Γ q . p F { A q is an isomorphism. Definition A lax morphism F : C Ñ D of natural models that preserves context extension and terminal objects is called a morphism of natural models . Zwanziger (CMU) Natural Models ˆ Comonads 15 / 31
Comonadic Type Theory Introduction 1 Natural Model Theory 2 Objects Morphisms Comonadic Type Theory 3 Semantics of Comonadic Type Theory 4 Cartesian Comonads on Natural Models Interpretation Zwanziger (CMU) Natural Models ˆ Comonads 16 / 31
Comonadic Type Theory CoTT Contexts and Judgments We will use the comonadic fragment of Shulman (2018)’s real-cohesive type theory. We have two variable judgments, denoted u :: A and x : A , and the typing judgement has form u 1 :: A 1 , ..., u m :: A m | x 1 : B 1 , ..., x n : B n $ t : C . Zwanziger (CMU) Natural Models ˆ Comonads 17 / 31
Comonadic Type Theory CoTT Contexts and Judgments (cont’d) The two variable judgements lead to a duplication of the context and variable rules: Emp. ¨ | ¨ $ ∆ , u :: A , ∆ 1 | Γ $ ∆ | ¨ $ B type Ext. 5 Var. 5 ∆ , u :: A , ∆ 1 | Γ $ u : A ∆ , u :: B | ¨ $ Zwanziger (CMU) Natural Models ˆ Comonads 18 / 31
Comonadic Type Theory CoTT Contexts and Judgments (cont’d) The two variable judgements lead to a duplication of the context and variable rules: Emp. ¨ | ¨ $ ∆ , u :: A , ∆ 1 | Γ $ ∆ | ¨ $ B type Ext. 5 Var. 5 ∆ , u :: A , ∆ 1 | Γ $ u : A ∆ , u :: B | ¨ $ ∆ | Γ , x : A , Γ 1 $ ∆ | Γ $ B type Ext. Var. ∆ | Γ , x : A , Γ 1 $ x : A ∆ | Γ , x : B $ Zwanziger (CMU) Natural Models ˆ Comonads 18 / 31
Comonadic Type Theory CoTT The Comonad 5 ∆ | ¨ $ t : B ∆ | ¨ $ B type 5 -Intro. 5 -Form. ∆ | Γ $ t 5 : 5 B ∆ | Γ $ 5 B type Zwanziger (CMU) Natural Models ˆ Comonads 19 / 31
Comonadic Type Theory CoTT The Comonad 5 ∆ | ¨ $ t : B ∆ | ¨ $ B type 5 -Intro. 5 -Form. ∆ | Γ $ t 5 : 5 B ∆ | Γ $ 5 B type ∆ | Γ , x : 5 A $ B type ∆ , u :: A | Γ $ t : B r u 5 { x s ∆ | Γ $ s : 5 A 5 -Elim. ∆ | Γ $ p let u 5 : “ s in t q : B r s { x s Zwanziger (CMU) Natural Models ˆ Comonads 19 / 31
Comonadic Type Theory CoTT The Comonad 5 (Conversions) ∆ | Γ , x : 5 A $ B type ∆ , u :: A | Γ $ t : B r u 5 { x s ∆ | ¨ $ s : A 5 - β -Conv. ∆ | Γ $ p let u 5 : “ s 5 in t q ” t r s { u s : B r s 5 { x s ∆ | Γ , x : 5 A $ B type ∆ | Γ $ s : 5 A ∆ | Γ , x : 5 A $ t : B 5 - η -Conv. ∆ | Γ $ let u 5 : “ s in t r u 5 { x s ” t r s { x s : B r s { x s Zwanziger (CMU) Natural Models ˆ Comonads 20 / 31
Semantics of Comonadic Type Theory Cartesian Comonads on Natural Models Introduction 1 Natural Model Theory 2 Objects Morphisms Comonadic Type Theory 3 Semantics of Comonadic Type Theory 4 Cartesian Comonads on Natural Models Interpretation Zwanziger (CMU) Natural Models ˆ Comonads 21 / 31
Semantics of Comonadic Type Theory Cartesian Comonads on Natural Models Cartesian Comonads Our notion of model for CoTT takes an appealingly simple form: Zwanziger (CMU) Natural Models ˆ Comonads 22 / 31
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