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A General Framework for the Semantics of TypeTheory
Taichi Uemura
ILLC, University of Amsterdam
9 July, 2019. CT
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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.
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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.
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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 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.
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CwF vs Natural Model
The representable map p : E→ U models context comprehension:
よ{A} E
よΓ U
δA
πAy
p
A
よ{A} ∼= A∗E
Proposition (Awodey 2018)
CwFs ' natural models.
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CwF vs Natural Model
The representable map p : E→ U models context comprehension:
よ{A} E
よΓ U
δA
πAy
p
A
よ{A} ∼= A∗E
Proposition (Awodey 2018)
CwFs ' natural models.
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Modeling Type Formers
Dependent function types (Π-types) are modeled by a pullback
PpE E
PpU U
λ
Pppy
p
Π
where Pp : [Sop, Set]→ [Sop, Set] is the functor
[Sop, Set] [Sop,Set]/E [Sop, Set]/U [Sop, Set](−×E) p∗ dom
and p∗ is the pushforward along p, i.e. the right adjoint of thepullback p∗.
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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;
the pushforward f∗ : A/X→ A/Y along a representable arrowf : X→ Y exists.
Definition
A representable map functor F : A→ B between representable mapcategories is a functor F : A→ B preserving all structures:representable arrows; finite limits; pushforwards along representablearrows.
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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;
the pushforward f∗ : A/X→ A/Y along a representable arrowf : X→ Y exists.
Definition
A representable map functor F : A→ B between representable mapcategories is a functor F : A→ B preserving all structures:representable arrows; finite limits; pushforwards along representablearrows.
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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 representable map functor (−)S : T→ [Sop, Set].
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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 representable map functor (−)S : T→ [Sop, Set].
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Examples of Type Theories
Proposition
Representable map categories have some “free” constructions (cf.LCCCs and Martin-Lof type theories (Seely 1984)).
Example
If T is freely generated by a single representable arrow p : E→ U,a model of T consists of...
a category S with a terminal object;
a representable map pS : ES → US of presheaves over S
i.e. a natural model.
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Examples of Type Theories
Proposition
Representable map categories have some “free” constructions (cf.LCCCs and Martin-Lof type theories (Seely 1984)).
Example
If T is freely generated by a single representable arrow p : E→ U,a model of T consists of...
a category S with a terminal object;
a representable map pS : ES → US of presheaves over S
i.e. a natural model.
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Outline
1 Introduction
2 Natural Models
3 Type Theories
4 Semantics of Type Theories
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Main Results
Let T be a type theory.
Theorem
The 2-category ModT of models of T has a bi-initial object.
Theorem
There is a “theory-model correspondence”: we define a (locallydiscrete) 2-category ThT of T-theories and establish a bi-adjunction
ModT
a
ThT.
LT
MT
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Main Results
Let T be a type theory.
Theorem
The 2-category ModT of models of T has a bi-initial object.
Theorem
There is a “theory-model correspondence”: we define a (locallydiscrete) 2-category ThT of T-theories and establish a bi-adjunction
ModT
a
ThT.
LT
MT
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Main Results
Let T be a type theory.
Theorem
The 2-category ModT of models of T has a bi-initial object.
Theorem
There is a “theory-model correspondence”: we define a (locallydiscrete) 2-category ThT of T-theories and establish a bi-adjunction
ModT
a
ThT.
LT
MT
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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.
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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.
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Internal Languages
Definition
We define a 2-functor LT : ModT → Cart(T,Set) byLTS(A) = A
S(1), where Cart(T,Set) is the category of functorsT→ Set preserving finite limits.
Theorem
LT : ModT → Cart(T, Set) has a left bi-adjoint with invertible unit.
ThT := Cart(T,Set)
(Cf. algebraic approaches to dependent type theory (Isaev 2018;Garner 2015; Voevodsky 2014))
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Internal Languages
Definition
We define a 2-functor LT : ModT → Cart(T,Set) byLTS(A) = A
S(1), where Cart(T,Set) is the category of functorsT→ Set preserving finite limits.
Theorem
LT : ModT → Cart(T, Set) has a left bi-adjoint with invertible unit.
ThT := Cart(T,Set)
(Cf. algebraic approaches to dependent type theory (Isaev 2018;Garner 2015; Voevodsky 2014))
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Internal Languages
Definition
We define a 2-functor LT : ModT → Cart(T,Set) byLTS(A) = A
S(1), where Cart(T,Set) is the category of functorsT→ Set preserving finite limits.
Theorem
LT : ModT → Cart(T, Set) has a left bi-adjoint with invertible unit.
ThT := Cart(T,Set)
(Cf. algebraic approaches to dependent type theory (Isaev 2018;Garner 2015; Voevodsky 2014))
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Conclusion
A type theory is a representable map category.
Every type theory has a bi-initial model.
There is a theory-model correspondence.
Future Directions:
Application: canonicity by gluing representable mapcategories?
What can we say about the 2-categoty ModT?
Better presentations of the category ThT?
Variations: internal type theories? (∞, 1)-type theories?
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References I
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.
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.
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References II
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.
R. Garner (2015). “Combinatorial structure of type dependency”.In: Journal of Pure and Applied Algebra 219.6, pp. 1885–1914.doi: 10.1016/j.jpaa.2014.07.015.
V. Isaev (2018). Algebraic Presentations of Dependent TypeTheories. arXiv: 1602.08504v3.
R. A. G. Seely (1984). “Locally cartesian closed categories andtype theory”. In: Math. Proc. Cambridge Philos. Soc. 95.1,pp. 33–48. doi: 10.1017/S0305004100061284.
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.
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References III
V. Voevodsky (2014). B-systems. arXiv: 1410.5389v1.
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Why is it a Theory?
In algebraic approaches to dependent type theory (Isaev 2018;Garner 2015; Voevodsky 2014), a theory is a diagram in Set whichlooks like
E0 E1 E2 . . .
U0 U1 U2 . . .
where
Un set of types with n variables;
En set of terms with n variables.
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Why is it a Theory?
If T has a representable arrow p : E→ U, then T contains adiagram
P0pE P1
pE P2pE . . .
P0pU P1
pU P2pU . . .
P0pp P1
pp P2pp
where PpX = p∗(X× E).