A General Framework for the Semantics of TypeTheory

Taichi Uemura

ILLC, University of Amsterdam

16 August, 2019. HoTT 2019

CwF-semantics of Type Theory

Semantics of type theories based on categories with families (CwF)(Dybjer 1996).

Martin-Lof type theory

Homotopy type theory

Homotopy type system (Voevodsky 2013) and two-level typetheory (Annenkov, Capriotti, and Kraus 2017)

Cubical type theory (Cohen et al. 2018)

Goal

To define a general notion of a “type theory” to unify theCwF-semantics of various type theories.

Outline

1 Introduction

2 Natural Models

3 Type Theories

4 Semantics of Type Theories

Outline

1 Introduction

2 Natural Models

3 Type Theories

4 Semantics of Type Theories

Natural Models

An alternative definition of CwF.

Definition (Awodey 2018)

A natural model consists of...

a category S (with a terminal object);

a map p : E→ U of presheaves over S

such that p is representable: for any object Γ ∈ S and elementA ∈ U(Γ), the presheaf A∗E defined by the pullback

A∗E E

よΓ U

yp

A

is representable, where よ is the Yoneda embedding.

Interpreting Type Theory

Natural model Type theory

Γ ∈ S Γ ` ctxA ∈ U(Γ) Γ ` A type

a ∈ {x ∈ E(Γ) | p(x) = A} Γ ` a : Af : ∆→ Γ context morphismA · f ∈ U(∆) substitutiona · f ∈ E(∆) substitution

Representable Maps

The representable map p : E→ U models context comprehension:

よ{A} E

よΓ U

δA

πAy

p

A

よ{A} ∼= A∗E

Natural model Type theory

A :よΓ → U Γ ` A type

{A} ∈ S Γ , x : A ` ctxπA : {A}→ Γ (Γ , x : A)→ Γ

A · πA :よ{A}→ U Γ , x : A ` A type

δA :よ{A}→ E Γ , x : A ` x : A

Variable Binding

Variable binding is modeled by the pushforwardp∗ : [S

op,Set]/E→ [Sop, Set]/U, that is, the right adjoint to thepullback p∗.

Example

p∗(E×U) is the presheaf of type families: for Γ ∈ S andA :よΓ → U, we have

p∗(E×U)

よΓ UA

∼=

{よ{A} U

},

so a section of p∗(E×U) over A is a type family Γ , x : A ` B type.

Modeling Type Constructors

Consider dependent function types (Π-types).

Γ ` A type Γ , x : A ` B type

Γ `∏x:A

B type

It is modeled by an operation Π such that

ΠΓ (A,B) ∈ U(Γ) for Γ ∈ S, A ∈ U(Γ) and B ∈ U({A});Π commutes with substitution.

Thus Π is a map p∗(E×U)→ U of presheaves.

Cubical Type Theory

To model (cartesian) cubical type theory, we need morerepresentable maps.

Example

Contexts can be extended by an interval:

Γ ` ctxΓ , i : I ` ctx

This is modeled by a presheaf I such that the map I→ 1 isrepresentable.

Summary on Natural Models

An (extended) natural model consists of...

a category S (with a terminal object);

some presheaves U,E, . . . over S;

some representable maps p : E→ U, . . .;

some maps X→ Y of presheaves over S where X and Y arebuilt up from U,E, . . . ,p, . . . using finite limits andpushforwards along the representable maps p, . . ..

Outline

1 Introduction

2 Natural Models

3 Type Theories

4 Semantics of Type Theories

Representable Map Categories

Definition

A representable map category is a category A equipped with aclass of arrows called representable arrows satisfying the following:

A has finite limits;

identity arrows are representable and representable arrows areclosed under composition;

representable arrows are stable under pullbacks;

representable arrows are exponentiable: the pushforwardf∗ : A/X→ A/Y along a representable arrow f : X→ Y exists.

Example

[Sop,Set] with representable maps of presheaves.

Representable Map Categories

Proposition (Weber 2015)

Exponentiable arrows are stable under pullbacks.

Example

A category A with finite limits has structures of a representablemap category:

Smallest one only isomorphisms are representable;

Largest one all exponentiable arrows are representable.

Also, given a class R of exponentiable arrows, we have the smalleststructure of a representable map category containing R.

Type Theories

Definition

A type theory is a (small) representable map category T.

Definition

A model of a type theory T consists of...

a category S with a terminal object;

a morphism (−)S : T→ [Sop,Set] of representable mapcategories, i.e. a functor preserving everything.

Cf. Functorial semantics of algebraic theories (Lawvere 1963),first-order categorical logic (Makkai and Reyes 1977)

Generalised Algebraic Theories

We give an example G of a type theory whose models are preciselythe natural models.

Definition

We denote by G the opposite of the category of finitely presentablegeneralised algebraic theories (GATs) (Cartmell 1978).

From the general theory of locally presentable categories (Adamekand Rosicky 1994), we get:

Proposition

G is essentially small and has finite limits, and Funfinlim(G,Set) isequivalent to the category of generalised algebraic theories.

An Exponentiable Map of GATs

Definition

U0 ∈ G is the GAT consisting of a type constant A0.

E0 ∈ G is the GAT consisting of a type constant A0 and aterm constant a0 : A0.

∂0 : E0 → U0 is the arrow in G represented by the inclusionU0 → E0.

Proposition

∂0 : E0 → U0 in G is exponentiable.

So G has the smallest structure of a representable map categorycontaining ∂0.

An Exponentiable Map of GATs

Example

Let Σ denote the finite GAT

` B type

x1 : B, x2 : B ` C(x1, x2) type

x : B ` c(x) : C(x, x).

Then (∂0)∗(E0 × Σ) is the finite GAT

` A0 type

x0 : A0 ` B(x0) type

x0 : A0, x1 : B(x0), x2 : B(x0) ` C(x0, x1, x2) type

x0 : A0, x : B(x0) ` c(x0, x) : C(x0, x, x).

Representable Map Category of Finite GATs

Theorem

G is “freely generated by ∂0” as a representable map category: fora representable map category A and a representable arrowf : X→ Y in A, there exists a unique, up to isomorphism,morphism F : G→ A of representable map categories equippedwith an isomorphism F∂0

∼= f.

Corollary

Models of G ' Natural models (' CwFs)

Outline

1 Introduction

2 Natural Models

3 Type Theories

4 Semantics of Type Theories

Bi-initial Models

Let T be a type theory.

Theorem

The 2-category ModT of models of T has a bi-initial object.

Theory-model Correspondence

Definition

A T-theory is a functor T→ Set preserving finite limits. PutThT := Funfinlim(T,Set).

Example

A G-theory is a generalised algebraic theory.

Theory-model Correspondence

Definition

We define the internal language 2-functor LT : ModT → ThT as

LT(S) =

(T [Sop, Set] Set

(−)S X 7→X(1))

.

Theorem

LT has a left bi-adjoint with invertible unit.

ModT

aThT.

LT

MT

Theory-model Correspondence

Example

When T = G, we get a bi-adjunction

CwFs

a

GATs

Conclusion

Definition

A type theory is a (small) representable map category T.

Further results and future directions:

Logical framework for representable map categories

Application: canonicity by gluing representable map categories(instead of gluing models)?

What can we say about the 2-categoty ModT?

What can we say about the category ThT?

Variations: internal type theories? (∞, 1)-type theories?

References I

J. Adamek and J. Rosicky (1994). Locally Presentable andAccessible Categories. Vol. 189. London Mathematical SocietyLecture Note Series. Cambridge University Press.

D. Annenkov, P. Capriotti, and N. Kraus (2017). Two-Level TypeTheory and Applications. arXiv: 1705.03307v2.

S. Awodey (2018). “Natural models of homotopy type theory”. In:Mathematical Structures in Computer Science 28.2,pp. 241–286. doi: 10.1017/S0960129516000268.

J.W. Cartmell (1978). “Generalised algebraic theories andcontextual categories”. PhD thesis. Oxford University.

References II

C. Cohen et al. (2018). “Cubical Type Theory: A ConstructiveInterpretation of the Univalence Axiom”. In: 21st InternationalConference on Types for Proofs and Programs (TYPES 2015).Ed. by T. Uustalu. Vol. 69. Leibniz International Proceedings inInformatics (LIPIcs). Dagstuhl, Germany: SchlossDagstuhl–Leibniz-Zentrum fuer Informatik, 5:1–5:34. doi:10.4230/LIPIcs.TYPES.2015.5.

P. Dybjer (1996). “Internal Type Theory”. In: Types for Proofs andPrograms: International Workshop, TYPES ’95 Torino, Italy,June 5–8, 1995 Selected Papers. Ed. by S. Berardi andM. Coppo. Berlin, Heidelberg: Springer Berlin Heidelberg,pp. 120–134. doi: 10.1007/3-540-61780-9_66.

F. W. Lawvere (1963). “Functorial Semantics of AlgebraicTheories”. PhD thesis. Columbia University.

References III

M. Makkai and G. E. Reyes (1977). First Order Categorical Logic.Model-Theoretical Methods in the Theory of Topoi and RelatedCategories. Vol. 611. Lecture Notes in Mathematics.Springer-Verlag Berlin Heidelberg. doi: 10.1007/BFb0066201.

V. Voevodsky (2013). A simple type system with two identitytypes. url: https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/HTS.pdf.

M. Weber (2015). “Polynomials in categories with pullbacks”. In:Theory and Applications of Categories 30.16, pp. 533–598.

The Bi-initial Model

For a type theory T, we define a model I(T) of T:

the base category is the full subcategory of T consisting ofthose Γ ∈ T such that the arrow Γ → 1 is representable;

we define (−)I(T) to be the composite

T よ−→ [Top,Set]→ [I(T)op,Set].

Given a model S of T, we have a functor

I(T) S

T [Sop,Set]

F

よ∼=

(−)S

and F can be extended to a morphism of models of T.