## Internal Type Theory (1996)

Venue: | Lecture Notes in Computer Science |

Citations: | 40 - 7 self |

### BibTeX

@INPROCEEDINGS{Dybjer96internaltype,

author = {Peter Dybjer},

title = {Internal Type Theory},

booktitle = {Lecture Notes in Computer Science},

year = {1996},

pages = {120--134},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

. We introduce categories with families as a new notion of model for a basic framework of dependent types. This notion is close to ordinary syntax and yet has a clean categorical description. We also present categories with families as a generalized algebraic theory. Then we define categories with families formally in Martin-Lof's intensional intuitionistic type theory. Finally, we discuss the coherence problem for these internal categories with families. 1 Introduction In a previous paper [8] I introduced a general notion of simultaneous inductiverecursive definition in intuitionistic type theory. This notion subsumes various reflection principles and seems to pave the way for a natural development of what could be called "internal type theory", that is, the construction of models of (fragments of) type theory in type theory, and more generally, the formalization of the metatheory of type theory in type theory. The present paper is a first investigation of such an internal type theor...

### Citations

281 |
Constructive mathematics and computer programming, in
- Martin-Löf
- 1982
(Show Context)
Citation Context ...y set-theoretic metalanguage. The setoid model of type theory can be viewed as a formalization of the standard "intuitive" model of type theory. It also justifies the rules of extensional ty=-=pe theory [16]-=- and certain rules for quotient formation. Our categorical approach to internal type theory can be contrasted to the syntactic approach by Pollack [19]. He formalized the syntax of dependent type theo... |

93 | Categorical logic
- Pitts
- 2000
(Show Context)
Citation Context ...propose how to solve it by constructing a model of normal forms. 2 Categories with Families 2.1 Basic Definitions Categories with families (cwfs) are variants of Cartmell's categories with attributess=-=[12, 18]-=-. The point of the reformulation is to get a more direct link to the syntax of dependent types. In particular we avoid reference to pullbacks, which give rise to a conditional equation when formalized... |

69 | The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions
- Pollack
- 1994
(Show Context)
Citation Context ...o justifies the rules of extensional type theory [16] and certain rules for quotient formation. Our categorical approach to internal type theory can be contrasted to the syntactic approach by Pollack =-=[19]-=-. He formalized the syntax of dependent type theory in type theory and proved syntactic properties such as Church-Rosser. The plan of the paper is as follows. In section 2 we introduce categores with ... |

61 |
Locally cartesian closed categories and type theory
- Seely
- 1984
(Show Context)
Citation Context ...y [2]. 4 The Coherence Problem for Internal Cwfs Categorical interpretations of type theory can be divided into those which interpret type equality as isomorphism, such as Seely's lccc-interpretation =-=[21]-=-, and those which interpret it as true equality, such as Cartmell's category with attributes interpretation. In either case a coherence problem arises. In order to interpret syntax in an lccc we have ... |

60 |
Generalised algebraic theories and contextual categories, Ann
- Cartmell
- 1986
(Show Context)
Citation Context ...where preservation is up to isomorphism. 2.2 The Generalized Algebraic Theory of Categories with Families The next step is to formalize cwfs as a generalized algebraic theory in the sense of Cartmell =-=[5]-=-. Generalized algebraic theories generalize many-sorted algebraic theories and are based on a framework of dependent types. They have four parts: -- A list of sort symbols with dependent typings. -- A... |

51 |
Extensional concepts in intensional type theory
- HOFMANN
- 1995
(Show Context)
Citation Context ...es with families to model a basic framework of dependent types and show how to formalize them in intensional intuitionistic type theory. One goal is to represent Hofmann's setoid model of type theory =-=[11, 13]-=- in type theory. He also used a categorical notion of model of dependent types (Cartmell's categories with attributes) but worked in ordinary set-theoretic metalanguage. The setoid model of type theor... |

47 | Intuitionistic model constructions and normalization proofs
- Coquand, Dybjer
- 1997
(Show Context)
Citation Context ...e lower an inclusion of normal forms. The two crucial properties of normalization are (i) that two convertible terms have identical normal forms and (ii) that a term is convertible to its normal form =-=[6]-=-. In our case property (i) is a consequence of the fact that equality in N is the basic I-equality in type theory. Property (ii) is a consequence of the equivalence of D and N and can be expressed as ... |

44 | On the interpretation of type theory in locally cartesian closed categories
- Hofmann
- 1994
(Show Context)
Citation Context ...propose how to solve it by constructing a model of normal forms. 2 Categories with Families 2.1 Basic Definitions Categories with families (cwfs) are variants of Cartmell's categories with attributess=-=[12, 18]-=-. The point of the reformulation is to get a more direct link to the syntax of dependent types. In particular we avoid reference to pullbacks, which give rise to a conditional equation when formalized... |

42 | Syntax and semantics of dependent types
- Hofmann
- 1997
(Show Context)
Citation Context ...rkshop on Categories and Type Theory, Goteborg, January 1995 [9]. Much useful information on cwfs can also be found in the lecture notes on "Syntax and Semantics of Dependent Types" by Marti=-=n Hofmann [14]-=-. He uses ordinary set-theoretic cwfs as the central semantic notion and gives several examples. He also discusses the relationship with other categorical notions of model for dependent types and give... |

40 |
Fibered categories and the foundations of naive category theory
- Bénabou
- 1985
(Show Context)
Citation Context ...traightforward [13]. But here a coherence problem arises when one already has an lccc and wants to construct a category with attributes. For this purpose Hofmann [12] adapted a method due to B'enabou =-=[3]-=- for constructing a split fibration from an arbitrary fibration. There is also a coherence problem for internal type theory. Proofs of type equality appear in terms and it is sometimes essential to kn... |

28 | Constructive category theory
- Huet, Säıbi
- 2000
(Show Context)
Citation Context ...ym (with an italic letter as a subscript) stand for the proofs of reflexivity, transitivity, and symmetry, respectively. Categories and Functors in Type Theory. We follow Aczel [1] and Huet and Saibi =-=[15]-=- and define a category to have a set of objects, but hom-setoids. We shall not need to refer to equality of objects. The object part of a functor is a function between the object sets and the morphism... |

21 |
Categorical Abstract Machines for Higher- Order Typed Lambda Calculi
- Ritter
- 1992
(Show Context)
Citation Context ...y, since our base category has a set and not a setoid of objects (contexts). We note the similarity to Martin-Lof's substitution calculus [17] which (unlike Ehrhard's [10], Curien's [7], and Ritter's =-=[20]-=-) lacks a judgement for context equality. The element constructors in the definition can be divided into three kinds: -- Those which correspond to operator symbols of the generalized algebraic theory ... |

18 |
Substitutions up to isomorphisms
- Curien
- 1993
(Show Context)
Citation Context ...or context equality, since our base category has a set and not a setoid of objects (contexts). We note the similarity to Martin-Lof's substitution calculus [17] which (unlike Ehrhard's [10], Curien's =-=[7]-=-, and Ritter's [20]) lacks a judgement for context equality. The element constructors in the definition can be divided into three kinds: -- Those which correspond to operator symbols of the generalize... |

17 | Extracting a proof of coherence for monoidal categories from a proof of normalization for monoids
- Beylin, Dybjer
- 1996
(Show Context)
Citation Context ...antics for Type Theory" by Per Martin-Lof (lecture at the meeting Twenty-Five Years of Constructive Type Theory", Venice, October, 1995). The reader is also referred to the paper by Beylin a=-=nd Dybjer [4]-=- which shows how related phenomena appear in another proof of coherence in type theory. Acknowledgements. The author is grateful for support from the ESPRIT BRA's TYPES and CLICS-II, from TFR (the Swe... |

9 |
Universes and a general notion of simultaneous inductive–recursive de nition in type theory
- Dybjer
- 1992
(Show Context)
Citation Context ...ith families formally in Martin-Lof's intensional intuitionistic type theory. Finally, we discuss the coherence problem for these internal categories with families. 1 Introduction In a previous paper =-=[8]-=- I introduced a general notion of simultaneous inductiverecursive definition in intuitionistic type theory. This notion subsumes various reflection principles and seems to pave the way for a natural d... |

8 |
Galois - a theory development project. Report for the 1993 Turin meeting on the Representation of Mathematics in Logical Frameworks. Available at URL: www.cs.man.ac.uk/˜petera/papers.html
- Aczel
(Show Context)
Citation Context ...rier, and ref, trans, sym (with an italic letter as a subscript) stand for the proofs of reflexivity, transitivity, and symmetry, respectively. Categories and Functors in Type Theory. We follow Aczel =-=[1]-=- and Huet and Saibi [15] and define a category to have a set of objects, but hom-setoids. We shall not need to refer to equality of objects. The object part of a functor is a function between the obje... |

8 |
A user's guide to
- Altenkirch, Gaspes, et al.
- 1994
(Show Context)
Citation Context ...we get an internal cwf N by interpreting equality of types, terms, and substitutions as I-equality. This result has been implemented in ALF, a proof checker for intensional intuitionistic type theory =-=[2]-=-. 4 The Coherence Problem for Internal Cwfs Categorical interpretations of type theory can be divided into those which interpret type equality as isomorphism, such as Seely's lccc-interpretation [21],... |

8 |
Elimination of extensionality and quotient types in martin-löf’s type theory
- Hofmann
- 1994
(Show Context)
Citation Context ...es with families to model a basic framework of dependent types and show how to formalize them in intensional intuitionistic type theory. One goal is to represent Hofmann's setoid model of type theory =-=[11, 13]-=- in type theory. He also used a categorical notion of model of dependent types (Cartmell's categories with attributes) but worked in ordinary set-theoretic metalanguage. The setoid model of type theor... |

6 |
Une s'emantique cat'egorique des types d'ependants. Application au calcul des constructions. Th`ese de doctorat, Universit'e de Paris VII
- Ehrhard
- 1988
(Show Context)
Citation Context ...t constructor for context equality, since our base category has a set and not a setoid of objects (contexts). We note the similarity to Martin-Lof's substitution calculus [17] which (unlike Ehrhard's =-=[10]-=-, Curien's [7], and Ritter's [20]) lacks a judgement for context equality. The element constructors in the definition can be divided into three kinds: -- Those which correspond to operator symbols of ... |

4 |
Substitution calculus. Notes from a lecture given in Goteborg
- Martin-Lof
- 1992
(Show Context)
Citation Context ...on when formalized in a straightforward way. Cwfs can therefore directly be formalized as a generalized algebraic theory with clear similiarities to Martin-Lof's substitution calculus for type theory =-=[17]-=-. Let Fam be the category of families of sets. An object is a family of sets (B(x)) x2A and a morphism with source (B(x)) x2A and target (B 0 (x 0 )) x 0 2A 0 is a pair consisting of a function f : A ... |