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On the Interpretation of Type Theory in Locally Cartesian Closed Categories
 Proceedings of Computer Science Logic, Lecture Notes in Computer Science
, 1994
"... . 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 exten ..."
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Cited by 38 (1 self)
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. 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 \Pitypes 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 ...
Developing Theories of Types and Computability via Realizability
, 2000
"... We investigate the development of theories of types and computability via realizability. ..."
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Cited by 20 (6 self)
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We investigate the development of theories of types and computability via realizability.
Reflexive Graphs and Parametric Polymorphism
, 1993
"... this paper is to understand why that is a parametric categorical model. In [10] Ma and Reynolds propose a parametricity hypothesis for a functor between categorical models of polymorphism which essentially requires that there is an extension of (a certain form of) an identity relation functor which ..."
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Cited by 18 (0 self)
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this paper is to understand why that is a parametric categorical model. In [10] Ma and Reynolds propose a parametricity hypothesis for a functor between categorical models of polymorphism which essentially requires that there is an extension of (a certain form of) an identity relation functor which preserve the model structure. There is no mention in the paper of any case when the parametricity hypothesis is satified, nor if there is a canonical completion of a category to one which satisfies the hypothesis. We shall suggest how the construction of a PLcategory of relations on a given category presented in [10] can be viewed as a "parametric completion". We shall also follow the suggestion of Ma in [9] that subtyping is a kind of parametricity requirement and show how to fit subtyping in the same setup. The basic idea is to use reflexive graphs of categories as in [12]. We shall employ their construction to present a kind of parametric completion of a given category. We also give a different presentation of the RELconstruction in [10], and use it to discuss some examples. We show in particular that the RELconstruction acts (essentially) in the same way on a category and on its completion. Hence it follows that the identity functor on the completion satisfies the parametricity hypothesis. Discussions with Eugenio Moggi, Peter O'Hearn, Edmund Robinson, and Thomas Streicher were very useful. Paul Taylor's beutiful diagram macros were used for typesetting all the diagrams in the text. 1 Graphs of categories
Exact Completions and Toposes
 University of Edinburgh
, 2000
"... Toposes and quasitoposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and ..."
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Cited by 13 (4 self)
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Toposes and quasitoposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and many of the latter arise by adding "good " quotients of equivalence relations to a simple category with finite limits. This construction is called the exact completion of the original category. Exact completions are not always toposes and it was not known, not even in the realizability and presheaf cases, when or why toposes arise in this way. Exact completions can be obtained as the composition of two related constructions. The first one assigns to a category with finite limits, the "best " regular category (called its regular completion) that embeds it. The second assigns to
Profunctors, open maps and bisimulation
 Mathematical Structures in Computer Science, To appear. Available from the Glynn Winskelâ€™s web
, 2000
"... ..."
Inductive Types and Exact Completion
 Ann. Pure Appl. Logic
, 2002
"... Using the theory of exact completions, we show that a specific class of pretopoi, consisting of what we might call "realizability pretopoi", can act as categorical models of certain predicative type theories, including MartinLof type theory. Our main theoretical instrument for doing so is a categor ..."
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Cited by 8 (7 self)
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Using the theory of exact completions, we show that a specific class of pretopoi, consisting of what we might call "realizability pretopoi", can act as categorical models of certain predicative type theories, including MartinLof type theory. Our main theoretical instrument for doing so is a categorical notion, the notion of weak Wtypes, an "intensional" analogue of the "extensional " notion of Wtypes introduced in an article by Moerdijk and Palmgren ([6]). 1
Type Theory via Exact Categories (Extended Abstract)
 In Proceedings of the 13th Annual IEEE Symposium on Logic in Computer Science LICS '98
, 1998
"... Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a categorytheoretic explanation of why ..."
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Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a categorytheoretic explanation of why the construction of a category of partial equivalence relations often produces a cartesian closed category. We show how several familiar examples of categories of partial equivalence relations fit into the general framework. 1 Introduction Partial equivalence relations (and categories of these) are a standard tool in semantics of programming languages, see e.g. [2, 5, 7, 9, 15, 17, 20, 22, 35] and [6, 29] for extensive surveys. They are usefully applied to give proofs of correctness and adequacy since they often provide a cartesian closed category with additional properties. Take for instance a partial equivalence relation on the set of natural numbers: a binary relation R ` N\ThetaN on th...
The Extensive Completion Of A Distributive Category
 Theory Appl. Categ
, 2001
"... A category with finite products and finite coproducts is said to be distributive if the canonical map AB+AC # A (B +C) is invertible for all objects A, B, and C. Given a distributive category D , we describe a universal functor D # D ex preserving finite products and finite coproducts, for wh ..."
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Cited by 7 (1 self)
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A category with finite products and finite coproducts is said to be distributive if the canonical map AB+AC # A (B +C) is invertible for all objects A, B, and C. Given a distributive category D , we describe a universal functor D # D ex preserving finite products and finite coproducts, for which D ex is extensive; that is, for all objects A and B the functor D ex /A D ex /B # D ex /(A + B) is an equivalence of categories. As an application, we show that a distributive category D has a full distributive embedding into the product of an extensive category with products and a distributive preorder. 1.
Maps II: Chasing Diagrams in Categorical Proof Theory
, 1996
"... In categorical proof theory, propositions and proofs are presented as objects and arrows in a category. It thus embodies the strong constructivist paradigms of propositionsastypes and proofsasconstructions, which lie in the foundation of computational logic. Moreover, in the categorical setting, ..."
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Cited by 7 (4 self)
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In categorical proof theory, propositions and proofs are presented as objects and arrows in a category. It thus embodies the strong constructivist paradigms of propositionsastypes and proofsasconstructions, which lie in the foundation of computational logic. Moreover, in the categorical setting, a third paradigm arises, not available elsewhere: logicaloperationsasadjunctions. It offers an answer to the notorious question of the equality of proofs. So we chase diagrams in algebra of proofs. On the basis of these ideas, the present paper investigates proof theory of regular logic: the f; 9gfragment of the first order logic with equality. The corresponding categorical structure is regular fibration. The examples include stable factorisations, sites, triposes. Regular logic is exactly what is needed to talk about maps, as total and singlevalued relations. However, when enriched with proofsasarrows, this familiar concept must be supplied with an additional conversion rule, conn...