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Categorical Models of Type Theory Theories and models Example Categorical models of type theory The theory of a group asserts an identity e, products x · y and inverses x−1 for any x; y, and equalities x · (y · z) = (x · y) · z and x · e = x = e · x and x · x−1 = e. Michael Shulman I A model of this theory (in sets) is a particularparticular group, like Z or S3. February 28, 2012 I A model in spaces is a topological group. I A model in manifolds is a Lie group. I ... 1 / 43 3 / 43 Group objects in categories Categorical semantics Definition A group object in a category with finite products is an object G with morphisms e : 1 ! G, m : G × G ! G, and i : G ! G, such that the following diagrams commute. Categorical semantics is a general procedure to go from 1. the theory of a group to m×1 (e;1) (1;e) G × G × G / G × G / G × G o 2. the notion of group object in a category. G F G FF xx 1×m m FF m xx A group object in a category is a model of the theory of a group. 1 FF xx1 FF# { xx G × G / x m G G Then, anything we can prove formally in the theory of a group will be valid for group objects in any category. ! / e / G 1 GO ∆ m G × G / G × G 1×i 4 / 43 5 / 43 Doctrines Theores and models For each kind of type theory there is a corresponding kind of structured category in which we consider models. Once we have fixed a doctrine D, then Algebraic theory ! Category with finite products I A D-theory specifies “generating” or “axiomatic” types and Simply typed λ-calculus ! Cartesian closed category terms. Dependent type theory ! Locally c.c. category I A D-category is one possessing the specified structure. I A model of a D-theory T in a D-category C realizes the types and terms in T as objects and morphisms of C. A doctrine specifies I A collection of type constructors (e.g. ×), and I A categorical structure realizing those constructors as operations (e.g. cartesian products). 6 / 43 7 / 43 The doctrine of finite products Models of finite-product theories T a finite-product theory, C a category with finite products. Definition Definition A finite-product theory is a type theory with unit and × as the A model of T in C assigns only type constructors, plus any number of axioms. 1. To each type A in T, an object A in C J K Example 2. To each judgment derivable in T: The theory of magmas has one axiomatic type M, and x : A ;:::; xn : An ` b : B axiomatic terms 1 1 a morphism in C: ` e : M and x : M; y : M ` (x · y): M b A1 × · · · × An −−!J K B : For monoids or groups, we need equality axioms (later). J K J K J K 3. Such that A × B = A × B , etc. J K J K J K 8 / 43 9 / 43 Models of finite-product theories Complete theories To define a model of T in C, it suffices to interpret the axioms. Example Definition A model of the theory of magmas in C consists of The complete theory Th(C) of a D-category C has I An object M . I As axiomatic types, all the objects of C. J K e I A morphism 1 −−!J K M . I As axiomatic terms, all the morphisms of C. J K · I A morphism M × M −!J K M . J K J K J K Remarks Given this, any other term like I The theory Th(C) has a tautological model in C. A model of T in C is equivalently a translation of T into x : M; y : M; z : M ` x · (y · z): M I Th(C). is automatically interpreted by the composite I Reasoning in Th(C), or a subtheory of it, is a way to prove things specifically about C. 1× · · M × M × M −−−!J K M × M −!J K M J K J K J K J K J K J K 10 / 43 11 / 43 Syntactic categories The syntax–semantics adjunction There are bijections between: Definition The syntactic category Syn(T) of a D-theory T has 1. Models of a theory T in a category C 2. Structure-preserving functors Syn(T) ! C I As objects, exactly the types of T. 3. Translations T ! Th(C) I As morphisms, exactly the terms of T. Hence Syn is left adjoint to Th. Remarks syntactic category I The theory T has a tautological model in Syn(T). A model of T in C is equivalently a structure-preserving I Type theories Categories functor Syn(T) ! C. I That is, Syn(T) ! C is the free D-category generated by a model of T. complete theory I Studying Syn(T) categorically can yield meta-theoretic information about T. Depending on how you set things up, you can make this adjunction an equivalence. 12 / 43 13 / 43 Why categorical semantics A list of doctrines unit ! terminal object I When we prove something in a particular type theory, like ; ! initial object the theory of a group, it is then automatically valid for product A × B ! categorical product models of that theory in all different categories. disjoint union A + B ! categorical coproduct I We can use type theory to prove things about a particular function type A ! B ! exponentials (cartesian closure) category by working in its complete theory. I We can use category theory to prove things about a type To include a type constructor in a doctrine, we have to specify theory by working with its syntactic category. meanings for 1. the type constructor (an operation on objects) 2. its constructors, and 3. its eliminators. 14 / 43 16 / 43 Universal properties Uniqueness of universal properties The categorical versions of type constructors are generally characterized by universal properties. Definition Theorem A left universal property for an object X of a category is a way If X and X 0 have the same universal property, then X =∼ X 0. of describing hom(X; Z ) up to isomorphism for every object Z , which is “natural in Z”. Example ∼ 0 ∼ Examples Suppose hom(;; Z ) = ∗ and hom(; ; Z) = ∗ for all Z . 0 0 I Then hom(;; ; ) =∼ ∗ and hom(; ; ;) =∼ ∗, so we have ∼ I hom(;; Z ) = ∗. morphisms ;!;0 and ;0 !;. hom(A + B; Z) ∼ hom(A; Z ) × hom(B; Z ). 0 0 I = I Also hom(;; ;) =∼ ∗ and hom(; ; ; ) =∼ ∗, so the composites ;!;0 !; and ;0 !;!;0 must be identities. Definition A right universal property for an object X of a category is a way of describing hom(Z; X) up to isomorphism for every object Z , which is “natural in Z”. 17 / 43 18 / 43 Interpreting positive types Interpreting positive types Positive type constructors are generally interpreted by objects Positive type constructors are generally interpreted by objects with left universal properties. with left universal properties. I The constructors are given as data along with the objects. I The constructors are given as data along with the objects. I The eliminators are obtained from the universal property. I The eliminators are obtained from the universal property. Example Example An initial object has hom(;; Z) =∼ ∗. A coproduct of A; B has morphisms inl: A ! A + B and I No extra data (no constructors). inr: B ! A + B, such that composition with inl and inr: I For every Z, we have a unique morphism ;! Z (the hom(A + B; Z ) ! hom(A; Z) × hom(B; Z) eliminator “abort” or “match with end”). is a bijection. I Two data inl and inr (type constructors of a disjoint union). I Given A ! Z and B ! Z, we have a unique morphism A + B ! Z (the eliminator, definition by cases). 19 / 43 19 / 43 Interpreting negative types Cartesian products are special Negative type constructors are generally interpreted by objects with right universal properties. I The eliminators are given as data along with the objects. Definition I The constructors are obtained from the universal property. A product of A; B has morphisms pr1 : A × B ! A and pr2 : A × B ! B, such that composition with pr1 and pr2: Example An exponential of A; B has a morphism ev: BA × A ! B, such hom(Z ; A × B) ! hom(Z ; A) × hom(Z ; B) that composition with ev: is a bijection. hom(Z; BA) ! hom(Z × A; B) I This is a right universal property. but we said products were a positive type! is a bijection. I Also: we already used products × in other places! I One datum ev (eliminator of function types, application). A I Given a morphism A ! B, we have a unique element of B (the constructor, λ-abstraction). 20 / 43 21 / 43 How to deal with products Display object categories Backing up: how do we interpret terms Definition A display object category is a category with x : A; y : B ` c : C I A terminal object. if we don’t have the type constructor ×? I A subclass of its objects called the display objects. (i.e. if our category of types doesn’t have products?) I The product of any object by a display object exists. 1. Work in a cartesian multicategory: in addition to Idea morphisms A ! C we have “multimorphisms” A; B ! C. I The objects represent contexts. 2. OR: associate objects to contexts rather than types. I The display objects represent singleton contexts x : A, which are equivalent to types. These are basically equivalent.
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