a general framework for the semantics of type theory · referencesi j. ad amek and j. rosick y...
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A General Framework for the Semantics of TypeTheory
Taichi Uemura
ILLC, University of Amsterdam
16 August, 2019. HoTT 2019
https://hott.github.io/HoTT-2019/
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CwF-semantics of Type Theory
Semantics of type theories based on categories with families (CwF)(Dybjer 1996).
Martin-Löf 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.
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Outline
1 Introduction
2 Natural Models
3 Type Theories
4 Semantics of Type Theories
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Outline
1 Introduction
2 Natural Models
3 Type Theories
4 Semantics of Type Theories
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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 Ssuch 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.
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Interpreting Type Theory
Natural model Type theory
Γ ∈ S Γ ` ctxA ∈ U(Γ) Γ ` A typea ∈ {x ∈ E(Γ) | p(x) = A} Γ ` a : Af : ∆→ Γ context morphismA · f ∈ U(∆) substitutiona · f ∈ E(∆) substitution
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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)→ ΓA · πA :よ{A}→ U Γ , x : A ` A typeδA :よ{A}→ E Γ , x : A ` x : A
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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.
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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.
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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.
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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, . . ..
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Outline
1 Introduction
2 Natural Models
3 Type Theories
4 Semantics of Type Theories
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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.
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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.
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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)
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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 (Adámekand Rosický 1994), we get:
Proposition
G is essentially small and has finite limits, and Funfinlim(G,Set) isequivalent to the category of generalised algebraic theories.
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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.
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An Exponentiable Map of GATs
Example
Let Σ denote the finite GAT
` B typex1 : B, x2 : B ` C(x1, x2) type
x : B ` c(x) : C(x, x).
Then (∂0)∗(E0 × Σ) is the finite GAT
` A0 typex0 : A0 ` B(x0) type
x0 : A0, x1 : B(x0), x2 : B(x0) ` C(x0, x1, x2) typex0 : A0, x : B(x0) ` c(x0, x) : C(x0, x, x).
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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)
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Outline
1 Introduction
2 Natural Models
3 Type Theories
4 Semantics of Type Theories
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Bi-initial Models
Let T be a type theory.
Theorem
The 2-category ModT of models of T has a bi-initial object.
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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.
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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
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Theory-model Correspondence
Example
When T = G, we get a bi-adjunction
CwFs
a
GATs
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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?
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References I
J. Adámek and J. Rosický (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.
https://arxiv.org/abs/1705.03307v2https://doi.org/10.1017/S0960129516000268
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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.
https://doi.org/10.4230/LIPIcs.TYPES.2015.5https://doi.org/10.1007/3-540-61780-9_66
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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.
https://doi.org/10.1007/BFb0066201https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/HTS.pdfhttps://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/HTS.pdf
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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.
IntroductionNatural ModelsType TheoriesSemantics of Type TheoriesConclusionReferences