## Formalizing categorical models of type theory in type theory (2007)

Venue: | In International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice |

Citations: | 5 - 2 self |

### BibTeX

@INPROCEEDINGS{Buisse07formalizingcategorical,

author = {Alexandre Buisse and Peter Dybjer},

title = {Formalizing categorical models of type theory in type theory},

booktitle = {In International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice},

year = {2007}

}

### OpenURL

### Abstract

This note is about work in progress on the topic of “internal type theory ” where we investigate the internal formalization of the categorical metatheory of constructive type theory in (an extension of) itself. The basic notion is that of a category with families, a categorical notion of model of dependent type theory. We discuss how to formalize the notion of category with families inside type theory and how to build initial categories with families. Initial categories with families will be term models which play the role of canonical syntax for dependent type theory. We also discuss the formalization of the result that categories with finite limits give rise to categories with families. This yields a type-theoretic perspective on Curien’s work on “substitution up to isomorphism”. Our formalization is being carried out in the proof assistant Agda 2 developed at Chalmers. 1

### Citations

88 | The HOL logic
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- 1993
(Show Context)
Citation Context ... In Section 6 we discuss the construction of initial categories with families. 2 Categories with families Categories with families (cwfs) [11,17] are variants of Cartmell’s categories with attributes =-=[16,21]-=-. 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... |

76 | Inductive sets and families in Martin-Löfs Type Theory and their set-theoretic semantics: An inversion principle for Martin-Löfs type theory
- Dybjer
- 1991
(Show Context)
Citation Context ...t it is a constructively meaningful definition. As part of our investigation we plan to generalize the schema in [10,15] to cover that schema, and also to provide set-theoretic semantics by extending =-=[9]-=-. 6.2 The category of cwfs in type theory. Although the above seems like a reasonable candidate for a strongly typed notion of term model of type theory, we would like to prove formally in type theory... |

68 | The theory of LEGO: A proof checker for the extended calculus of constructions
- Pollack
- 1994
(Show Context)
Citation Context ...o add certain analogues of large cardinals to constructive type theory. Previous work on constructive model theory which use a formal constructive metalanguage includes Pollack’s work on Lego in Lego =-=[22]-=- and Barras’ work on Coq in Coq [4]. Both authors deal with the usual lambda calculus based syntax of constructive type theory. Here we will instead base our work on a categorical notion of model of t... |

66 | Inductive families
- Dybjer
- 1994
(Show Context)
Citation Context ...txt data T ype : Ctxt → Set where . . . However, it is important to remark that this inductive definition falls outside the standard schema of mutual inductive definitions in constructive type theory =-=[10]-=-. Nevertheless, we believe that it is a constructively meaningful definition. As part of our investigation we plan to generalize the schema in [10,15] to cover that schema, and also to provide set-the... |

65 | A general formulation of simultaneous inductiverecursive definitions in type theory - Dybjer - 2000 |

58 |
Generalised algebraic theories and contextual categories. Annals of Pure and Applied Logic 32
- Cartmell
- 1986
(Show Context)
Citation Context ...he nose. Small cwfs and morphisms of cwfs form a category Cwf. As already mentioned the notion of a category with families can be formalized as a generalized algebraic theory in the sense of Cartmell =-=[6]-=-. It is instructive to look at the rules of this theory, but we do not have room in this short note to display them, and refer the reader to Dybjer [11]. 3 The proof assistant Agda 2 We will work in M... |

55 |
Locally cartesian closed categories and type theory
- Seely
- 1984
(Show Context)
Citation Context ... problems. After completing the formalizations described in this paper, we would like to add more structure to categories with families. In particular we would like to formalize the full Seely-Curien =-=[24,7]-=- interpretation of Martin-Löf type theory (understood as categories with families with extra structure modelling Π- and Σ-types and extensional equality types) in locally cartesian closed categories. ... |

43 | Indexed induction-recursion
- Dybjer, Setzer
- 2006
(Show Context)
Citation Context ...r.nl/locate/entcssBuisse and Dybjer of the inductive-recursive definitions which are needed in certain model constructions and normalization proofs. Although such definitions are constructively valid =-=[12,13,14,15]-=- most authors rely on their interpretation in set theory [23,3,1,2] and this also has formal repercussions. In this project we plan to show that it is possible to rely entirely on constructive notions... |

42 | A finite axiomatization of inductiverecursive definitions
- Dybjer, Setzer
- 1999
(Show Context)
Citation Context ...r.nl/locate/entcssBuisse and Dybjer of the inductive-recursive definitions which are needed in certain model constructions and normalization proofs. Although such definitions are constructively valid =-=[12,13,14,15]-=- most authors rely on their interpretation in set theory [23,3,1,2] and this also has formal repercussions. In this project we plan to show that it is possible to rely entirely on constructive notions... |

40 | Syntax and semantics of dependent types
- Hofmann
- 1996
(Show Context)
Citation Context ...y Hofmann [16] formulated using classical categorical notions. In Section 6 we discuss the construction of initial categories with families. 2 Categories with families Categories with families (cwfs) =-=[11,17]-=- are variants of Cartmell’s categories with attributes [16,21]. The point of the reformulation is to get a more direct link to the syntax of dependent types. In particular we avoid reference to pullba... |

38 |
Fibred categories and the foundations of naive category theory
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- 1985
(Show Context)
Citation Context ...rpreting type theory in locally cartesian closed categories [16]. In this work he shows how to construct a category with attributes from a category with finite limits using a technique due to Bénabou =-=[5]-=-. Since categories with attributes are equivalent to categories with families this ought to be highly relevant to our work. However, Hofmann uses standard category theory relying on set-theoretic meta... |

38 | On the interpretation of type theory in locally cartesian closed categories
- Hofmann
- 1994
(Show Context)
Citation Context ...he result that categories with finite limits give rise to cwfs. We discuss the close relationship to Curien’s paper “Substitution up to Isomorphism” [7] and contrast it to a similar result by Hofmann =-=[16]-=- formulated using classical categorical notions. In Section 6 we discuss the construction of initial categories with families. 2 Categories with families Categories with families (cwfs) [11,17] are va... |

36 | Internal type theory
- Dybjer
(Show Context)
Citation Context ...ategorical notion of model of type theory. The formal system of type theory will be represented abstractly as an initial category with families (with extra structure). A category with families (cwfs) =-=[11]-=- is a notion of model of (the most basic rules of) dependent type theory, and the initial such is the “term model”. There are several reasons for choosing a categorical approach. One of them is to ach... |

28 | Induction-recursion and initial algebras
- Dybjer, Setzer
- 2003
(Show Context)
Citation Context ...r.nl/locate/entcssBuisse and Dybjer of the inductive-recursive definitions which are needed in certain model constructions and normalization proofs. Although such definitions are constructively valid =-=[12,13,14,15]-=- most authors rely on their interpretation in set theory [23,3,1,2] and this also has formal repercussions. In this project we plan to show that it is possible to rely entirely on constructive notions... |

27 |
Auto-validation d’un système de preuves avec familles inductives. Thèse de doctorat
- Barras
- 1999
(Show Context)
Citation Context ...dinals to constructive type theory. Previous work on constructive model theory which use a formal constructive metalanguage includes Pollack’s work on Lego in Lego [22] and Barras’ work on Coq in Coq =-=[4]-=-. Both authors deal with the usual lambda calculus based syntax of constructive type theory. Here we will instead base our work on a categorical notion of model of type theory. The formal system of ty... |

25 | Constructive Category Theory
- Huet, Saïbi
- 1998
(Show Context)
Citation Context ... experience of formalization of category theory inside constructive type theory, see for example the development of elementary category theory in Huet and Saibi’s book on Constructive Category Theory =-=[18]-=-. Plan of the note. In Section 2 we present the notion of a cwf in classical metalanguage. In Section 3 we explain some of the features of the proof assistant Agda 2 which we use for our formalization... |

24 |
About models for intuitionistic type theories and the notion of definitional equality
- Martin-Löf
- 1975
(Show Context)
Citation Context ... repercussions. In this project we plan to show that it is possible to rely entirely on constructive notions on the metalevel. The idea of doing such constructive model theory goes back to Martin-Löf =-=[19]-=-. However, Martin-Löf relied on an informal constructive metalanguage, while we here are more specific and work with suitable versions of Martin-Löf type theory as formal metalanguages. We are checkin... |

22 | Normalization by evaluation for Martin-Löf type theory with one universe
- Abel, Aehlig, et al.
- 2007
(Show Context)
Citation Context ...ions which are needed in certain model constructions and normalization proofs. Although such definitions are constructively valid [12,13,14,15] most authors rely on their interpretation in set theory =-=[23,3,1,2]-=- and this also has formal repercussions. In this project we plan to show that it is possible to rely entirely on constructive notions on the metalevel. The idea of doing such constructive model theory... |

17 |
Substitution up to isomorphism
- Curien
- 1993
(Show Context)
Citation Context ...e theory. In Section 5 we sketch how to formalize the result that categories with finite limits give rise to cwfs. We discuss the close relationship to Curien’s paper “Substitution up to Isomorphism” =-=[7]-=- and contrast it to a similar result by Hofmann [16] formulated using classical categorical notions. In Section 6 we discuss the construction of initial categories with families. 2 Categories with fam... |

12 | A formalization of a dependently typed language as an inductive-recursive family
- Danielsson
- 2007
(Show Context)
Citation Context ...o formalize key metatheoretical results of Martin-Löf type theory such as decidability of equality and type-checking based on categories with families [1,2]. This is related to the work by Danielsson =-=[8]-=- who presented such a formalization of a normalization by evaluation result in the system AgdaLight, a precursor of the Agda 2 system. Danielsson did however not base his work on a categorical present... |

10 |
Combinators and classes
- Scott
(Show Context)
Citation Context ...ions which are needed in certain model constructions and normalization proofs. Although such definitions are constructively valid [12,13,14,15] most authors rely on their interpretation in set theory =-=[23,3,1,2]-=- and this also has formal repercussions. In this project we plan to show that it is possible to rely entirely on constructive notions on the metalevel. The idea of doing such constructive model theory... |

6 |
Frege Structures and the
- Aczel
- 1980
(Show Context)
Citation Context ...ions which are needed in certain model constructions and normalization proofs. Although such definitions are constructively valid [12,13,14,15] most authors rely on their interpretation in set theory =-=[23,3,1,2]-=- and this also has formal repercussions. In this project we plan to show that it is possible to rely entirely on constructive notions on the metalevel. The idea of doing such constructive model theory... |

4 |
Substitution calculus. Notes from a lecture given in Goteborg
- Martin-Lof
- 1992
(Show Context)
Citation Context ...malized in a straightforward way. Cwfs can therefore be formalized as a generalized algebraic theory in Cartmell’s sense with clear similiarities to Martin-Löf’s substitution calculus for type theory =-=[20]-=-. Let Fam be the category of families of sets, where an object is a family of sets (B(x))x∈A and a morphism with source (B(x))x∈A and target (B ′ (x ′ ))x ′ ∈A ′ is a pair consisting of a function f :... |

1 |
Normalization by evaluation for Martin-Lf type theory with equality judgements
- Abel, Coquand, et al.
- 2007
(Show Context)
Citation Context |