## On the Interpretation of Type Theory in Locally Cartesian Closed Categories (1994)

Venue: | Proceedings of Computer Science Logic, Lecture Notes in Computer Science |

Citations: | 39 - 1 self |

### BibTeX

@INPROCEEDINGS{Hofmann94onthe,

author = {Martin Hofmann},

title = {On the Interpretation of Type Theory in Locally Cartesian Closed Categories},

booktitle = {Proceedings of Computer Science Logic, Lecture Notes in Computer Science},

year = {1994},

pages = {427--441},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

. We show how to construct a model of dependent type theory (category with attributes) from a locally cartesian closed category (lccc). This allows to define a semantic function interpreting the syntax of type theory in an lccc. We sketch an application which gives rise to an interpretation of extensional type theory in intensional type theory. 1 Introduction and Motivation Interpreting dependent type theory in locally cartesian closed categories (lcccs) and more generally in (non split) fibrational models like the ones described in [7] is an intricate problem. The reason is that in order to interpret terms associated with substitution like pairing for \Sigma -types or application for \Pi-types one needs a semantical equivalent to syntactic substitution. To clarify the issue let us have a look at the "naive" approach described in Seely's seminal paper [14] which contains a subtle inaccuracy. Assume some dependently typed calculus like the one defined in [10] and an lccc C (a category ...

### Citations

394 |
Category Theory for Computing Science
- Barr, Wells
- 1999
(Show Context)
Citation Context ...it fibration due to Power which does not extend to \Pi- and \Sigma -types. Some familiarity with basic category theory and dependent type theory will be assumed. Introductory material may be found in =-=[1]-=- (categories) and [10] (dependent type theory). Both subjects are also well described in [12]. 2 Categories with attributes A category with attributes (cwa) is given by the following data -- a categor... |

261 |
Programming in Martin-Löf’s Type Theory: An Introduction
- Nordstrom, Petersson, et al.
- 1990
(Show Context)
Citation Context ...clarify the issue let us have a look at the "naive" approach described in Seely's seminal paper [14] which contains a subtle inaccuracy. Assume some dependently typed calculus like the one d=-=efined in [10]-=- and an lccc C (a category with finite limits and right adjoints to every pullback functor in order to interpret dependent product types.) The idea is to interpret contexts as objects in C, types in c... |

88 | The HOL logic
- Pitts
- 1993
(Show Context)
Citation Context ...ntactic and semantic substitution do agree. The technique of interpreting type theory in such a model has been worked out by Streicher [15] in great detail. See also Pitts' forthcoming survey article =-=[12]-=-. Unfortunately, however, it seems impossible to endow an arbitrary lccc with a pullback operation which would satisfy these coherence requirements. For example the natural choice of pullbacks in the ... |

58 |
Generalised algebraic theories and contextual categories. Annals of Pure and Applied Logic 32
- Cartmell
- 1986
(Show Context)
Citation Context ...ossible substitutions. More abstractly, we describe a construction which turns an arbitrary lccc into an equivalent category with attributes (cwa) --- a "split" notion of model introduced by=-= Cartmell [4]-=-, see also [12], for which an interpretation function is readily available. The method we use is a very general procedure due to B'enabou (see [2] and [7, Prop. 1.3.6]) which turns an arbitrary fibrat... |

56 |
Locally cartesian closed categories and type theory
- Seely
- 1984
(Show Context)
Citation Context ...gma -types or application for \Pi-types one needs a semantical equivalent to syntactic substitution. To clarify the issue let us have a look at the "naive" approach described in Seely's semi=-=nal paper [14]-=- which contains a subtle inaccuracy. Assume some dependently typed calculus like the one defined in [10] and an lccc C (a category with finite limits and right adjoints to every pullback functor in or... |

42 |
Categorical Type Theory
- Jacobs
- 1991
(Show Context)
Citation Context ...theory. 1 Introduction and Motivation Interpreting dependent type theory in locally cartesian closed categories (lcccs) and more generally in (non split) fibrational models like the ones described in =-=[7]-=- is an intricate problem. The reason is that in order to interpret terms associated with substitution like pairing for \Sigma -types or application for \Pi-types one needs a semantical equivalent to s... |

38 |
Fibred categories and the foundations of naive category theory
- Bénabou
- 1985
(Show Context)
Citation Context ... --- a "split" notion of model introduced by Cartmell [4], see also [12], for which an interpretation function is readily available. The method we use is a very general procedure due to B'en=-=abou (see [2]-=- and [7, Prop. 1.3.6]) which turns an arbitrary fibration into an equivalent split fibration. Our contribution consists of the observation that the cwa obtained thus has not merely a split substitutio... |

35 |
Some free constructions in realizability and proof theory
- Carboni
- 1995
(Show Context)
Citation Context ...t way. 6 Application: A category of setoids As mentioned in the Introduction for many lcccs an equivalent cwa is known already. However, there is an interesting example motivated by a construction in =-=[3]-=- for which the construction described in this paper seems to be the only viable way. Consider the syntax of intensional Martin-Lof type theory with natural numbers as described e.g. in [10]. We write ... |

23 | An introduction to fibrations, topos theory, the effective topos and modest sets. Lecture - Phoa |

18 |
A general coherence result
- Power
- 1989
(Show Context)
Citation Context .... The observation that the thus obtained cwa is closed under various type operators is to our knowledge original. Incidentally, for another somewhat dual construction of split fibrations due to Power =-=[13]-=- this is not the case. Using it Fam(\Gamma ) would be the set of pairs (s; oe) where s and oe are morphisms with common codomain and dom(s) = \Gamma . The associated canonical projection to such a fam... |

17 |
Substitution up to isomorphism
- Curien
- 1993
(Show Context)
Citation Context ... flaw of this idea is that the interpretation of �� [x := M ] is already fixed by the clauses of the interpretation and there is no reason why it should equal the chosen pullback of t along m. Cur=-=ien [5]-=- addresses the problem by making substitution a syntactic operator which may then be interpreted as (chosen) pullback. However, this changes the calculus and also results in a quite complicated interp... |

8 |
Semantics of Type Theory Birkhäuser
- Streicher
- 1991
(Show Context)
Citation Context ...semantic type and term formers. In this case one can show that syntactic and semantic substitution do agree. The technique of interpreting type theory in such a model has been worked out by Streicher =-=[15]-=- in great detail. See also Pitts' forthcoming survey article [12]. Unfortunately, however, it seems impossible to endow an arbitrary lccc with a pullback operation which would satisfy these coherence ... |

4 |
Comprehension categories and the semantics of type theory
- Jacobs
- 1993
(Show Context)
Citation Context ...dered this as an open problem. Locally cartesian closed categories play the role of a running example here; the arguments immediately carry over to the more general notions of model studied by Jacobs =-=[7, 8]-=- and other authors. On a more elementary level the method computes additional information along with the inductive definition of the interpretation which allows to identify the interpretation of a sub... |

1 |
Quotient types via coequalisers in Martin-Lof's type theory
- Mendler
(Show Context)
Citation Context ...ids is a worthwhile object for further study. In particular it appears to have coequalisers of equivalence relations and thus provides a model for the extensional quotient types studied by Mendler in =-=[9]-=-. Moreover, we believe that the full subcategory of the category of setoids consisting of those objects taken on by the interpretation function is actually equivalent to the lccc of types and terms in... |