J.M.E. Hyland and A.M. Pitts. The theory of constructions: Categorical semantics and topos-theoretic models. In Categories in Computer Science and Logic, volume 92 of Contemp. Math., pages 137--199, 1989. 161  Home/Search   Document Not in Database   Summary   Related Articles   Check

....type theories. Some of the earliest work in this area was undertaken by Cartmell [Car86] with additional work by Taylor [Tay86] Here we shall give a presentation of categories with attributes which is based on on Pitts account in [Pit95] Further useful information can be found in [Str89] [HP89], CGW89] and [Ben85] Categories with Attributes Definition 9.1.1 A category with attributes is specified by a category C with terminal object (called the base category) which is equipped with the following structure: ffl For each object X in C, a collection of fibrations over X, Fib(X) We ....

J.M.E. Hyland and A.M. Pitts. The theory of constructions: Categorical semantics and topos-theoretic models. In Categories in Computer Science and Logic, volume 92 of Contemp. Math., pages 137--199, 1989. 161

....N j n u(y) g is a well formed (subset) type in the context y : Y . For the technical development, we make use of B. Jacobs brational description of models of dependent type theory [23,25,26] which is related to the D categories [14] categories with attributes [12,30] display map categories [40,21], and comprehensive brations [32] See [23] for a comprehensive introduction. We make a point of describing the models in a so called split way, so as to avoid problems with interpreting dependent type theory. See, for example, 29,34,31,35,17] for a discussion of this issue. As this section ....

J.M.E. Hyland and A.M. Pitts. The theory of constructions: categorical semantics and topos theoretical models. In J.W. Gray and A. Scedrov, editors, Categories in Computer Science and Logic, volume 92 of Contemporary Mathematics, pages 137-199. AMS, Providence, RI,

....of constructive mathematics. 11, 14] CC may be viewed as the calculus associated, via the propositions as types principle [24] with natural deduction proofs in an extension of Church s higher order logic [6] The system has been proved both proof theoretically [11] and model theoretically [29, 17, 27] consistent, and the type checking problem has been proved decidable [14, 11] Although CC is an exceedingly rich formalism for expressing mathematical constructions, a variety of extensions to the calculus have been considered [12, 30, 31] These extensions are motivated by a variety of ....

J. Martin E. Hyland and Andrew M. Pitts. The Theory of Constructions: Categorical semantics and topos-theoretic models. In Proceedings of 45 the Boulder Conference on Categories in Computer Science, 1988. To appear.

....morphisms p and q as identities. There are various other notions of model all of which are essentially equivalent as far as interpretation of type theory in them is concerned. Locallycartesian closed categories (Seely 1984) and categories with display maps (Taylor 1986; Lamarche 1987; Hyland and Pitts 1989) are less general than CwFs because semantic types are identified with their associated projections. Usually, in these models the conditions corresponding to (Ty Comp) and (Ty Id) only hold up to isomorphism, which makes the definition of the interpretation function more complicated. See (Hofmann ....

Hyland, M. and A. Pitts (1989). The Theory of Constructions: Categorical Semantics and Topos-Theoretic Models. In Categories in Computer Science and Logic. AMS.

.... (in the second order case) This interpretation departs from Tarski s and it is much more insightful, yet still different from the one hinted here for geometry (see the Lawvere Grothendiek interpretation of quantification [Johnstone, 1977; Lambek Scott, 1989] and, for the second order case, [Hyland and Pitts, 1987; Asperti Longo,1991; Longo Moggi, 1991] 23 4.4.1 Proofs as Prototypes Following then the intuitionistic approach , let s try to introduce some basic ideas for a typetheoretic analysis of universal proofs, which stresses the role of invariance . Given a mathematical theory which allows ....

Hyland M., Pitts A. "The Theory of Constructions: categorical semantics and topos theoretic models" Categories in C.S. and Logic, Boulder (AMS notes), 1987.

....by [ C ] B Gamma Cat. A morphism (I ff Gamma C 0 ) Gamma (I fi Gamma C 0 ) in [I; C ] is a morphism I m Gamma C 1 such that the following diagram commutes. C 0 C 1 C 0 I m ff fi d 1 d 0 Let D be a collection of morphisms, called display maps ( Tay86, HP89] of B such that for any ffl d Gamma X in D and any Y f Gamma X, there is a (unique) morphism ffl c Gamma Y in D rendering the diagram Y X ffl ffl f c d a pullback. Let B=D be the full subcategory of the arrow category B determined by D. The definition of D simply says that ....

....language involves the build up of an internal definability relationship between the object language and the frame language. 28 10 The Notion of Models We now turn our attention to semantics. There are two levels of semantics. The model theory of TT has been well established, see for instance [HP89] A categorical model consists of a category B with finite products and a collection D of display maps. The fibration B=D cod Gamma B must be a fibred cartesian closed category and is complete relative to D. A generic judgement x : U Omega Gamma1 t U (x) must be interpreted as a display ....

J. Hyland and A. Pitts. The theory of constructions: Categorical semantics and topos-theoretic models. In J. Gray and A. Scedrov, editors, Categories in Computer Science and Logic, pages 137--199, Boulder, June 89. A.M.S.

....: T and : T 2 : T . So a strong monad is just a monad over C enriched over itself in the 2 category of C enriched categories. The second explanation was suggested to us by G. Plotkin, and takes as fundamental structure a class D of display maps over C, which models dependent types (see [2]) and induces a C indexed category C=D . Then a strong monad over a category C with nite products amounts to a monad over C=D in the 2 category of C indexed categories, where D is the class of rst projections (corresponding to constant type dependency) In general the natural transformation t ....

J.M.E. Hyland and A.M. Pitts. The theory of constructions: Categorical semantics and topostheoretic models. In Proc. AMS Conf. on Categories in Comp. Sci. and Logic (Boulder 1987), 1987.

....the category GC of HML modules. The Appendix gives a complete description of HML, while Section 7 investigates additional structure on the category GC. Section 6 reviews the category theoretic semantics of dependent types and the Calculus of Constructions (see [CH88] following quite closely [HP89], but using categories with attributes instead of classes of display maps. Section 7 investigates dependent kinds and dependent type schemas in HML, using the machinery set up in Section 6. More speci cally, it describes the e ect of independence of constructor expressions from ....

....are: Independence of type expressions from program expressions (as in system F ) enforced syntactically by having two levels of judgements (and contexts) so that HML can be viewed as an indexed category. Dependent kinds and type schemas (as in the Calculus of Constructions CC described in [HP89]) More precisely, HML has dependent sums and products for kinds and type schemas, and type schemas universally and existentially quanti ed over kinds. This enable us to analyse in full generality dependent types at the level of modules, that are necessary for giving a type theoretic account of ....

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J.M.E. Hyland and A.M. Pitts. The theory of constructions: Categorical semantics and topos-theoretic models. Contemporary Mathematics, 92, 1989.

....in [Laf88, See87] to model a fragment of linear logic) so that C can be enriched over itself. Then a strong monad over a cartesian closed category C is just a monad over C in the 2 category of C enriched categories. The second characterisation takes a class D of display maps over C (used in [HP87] to model dependent types) and de nes a C indexed category C=D . Then a strong monad over a category C with nite products amounts to a monad over C=D in the 2 category of C indexed categories, where D is the class of rst projections (corresponding to constant type dependency) In general the ....

J.M.E. Hyland and A.M. Pitts. The theory of constructions: Categorical semantics and topos-theoretic models. In Proc. AMS Conf. on Categories in Comp. Sci. and Logic (Boulder 1987), 1987.

.... intuitionistic setting are obtainable from the small complete internal poset Omega , whose object of objects is the subobject classifier [49, Ch.3] Subsequent research in categorical logic has considered the richer type systems above, with fibrations emerging as the central unifying concept [102,6,45, 26,105,84,48]. We shall return to these issues in Chapter 5 below. Type theoretically, the notion of subset , or subset type has been much less clearly defined. Probably the most closely argued and theoretically satisfying has been the work of the Goteborg group [79] One of the troubling (and desirable ) ....

....this harmony between the type system and the underlying untyped terms. Pavlovic, in his thesis [83] elaborates in categorical terms a theory of constructions in which programs do not depend on proofs of logical propositions. As with the models of Constructions considered by Hyland and Pitts [45], the emphasis is on extensional systems, rather than the intensional system we work with here. Proof theoretic properties seem to be regarded as something : an implementation would have to answer [83, p.8] Chapter 1. Introduction 14 1.5 Prerequisites We assume the reader is familiar ....

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J.M.E.Hyland and A.M.Pitts, The Theory of Constructions: Categorical Semantics and Topos-theoretic models, in: Proceedings of the AMS Conference on Categories in Computer Science, Boulder, Colorado, 1986.

....[5, 6, x14] had already put the propositions as types at work. The two facets of this paradigm were recognized in the very successfull type systems due to Martin Lof on one side [36] and to Girard [13] and Reynolds [44] on the other. Synthesis of these systems led to the Theory of Constructions [10, 21], the most comprehensive [40] type system so far. The categorical part of the story goes back to the sixties too. The first ideas of categorical logic were conceived in Lawvere s pursuit of foundations. Lambek s investigations in linguistics and algebra, on the other hand, led to a concrete ....

J.M.E. Hyland and A.M. Pitts, The Theory of Constructions: categorical semantics and topos-theoretic models in: J. Gray and A. Scedrov (eds.), Categories in Computer Science and Logic, Contemp. Math. 92 (Amer. Math. Soc. 1989)

.... [Luo90] This calculus is also the standard type theory used in the LEGO system [LP92] A related system is the specification language Gallina which is implemented in the Coq system [D 91] The semantics of CC has been studied by a number of authors from a categorical point of view, e.g. see [HP89] in [Ehr89] the notion of a dictos is introduced, and in [Jac91] the more general notion of a CC category is used. A very natural semantics based on the concept of Realizability is the set semantics. In [Str89] a mild generalization (D sets) is investigated in great detail and used to show ....

J.M.E. Hyland and A.M. Pitts. The theory of constructions: categorical semantics and topos-theoretic models. In J. Gray and A. Scedrov, editors, Categories in Computer Science and Logic, pages 137 -- 199, 1989.

....[5, 6, x14] had already put the propositions as types at work. The two facets of this paradigm were recognized in the very successful type systems due to Martin Lof on one side [36] and to Girard [13] and Reynolds [44] on the other. Synthesis of these systems led to the Theory of Constructions [10, 21], the most comprehensive [40] type system so far. The categorical part of the story goes back to the sixties too. The first ideas of categorical logic were conceived in Lawvere s pursuit of foundations. Lambek s investigations in linguistics and algebra, on the other hand, led to a concrete ....

J.M.E. Hyland and A.M. Pitts, The Theory of Constructions: categorical semantics and topostheoretic models in: J. Gray and A. Scedrov (eds.), Categories in Computer Science and Logic, Contemp. Math. 92 (Amer. Math. Soc. 1989)

.... Moggi s suggestion for a categorical understanding of Girard Troelstra models, by hinting the small completeness of a category of sets , several papers and discussions raised issues in Topos Theory and shed more light on topos theoretic models of Intuitionistic SetTheory (IZF; see Pitts[1987] Hyland[1987] and Asperti Longo[91] for detailed work and further references) At this point, one may wonder how constructive are the tools used in this kind of work. As the reader may see in the above references, the definition of the models requires the use of powerset operation and second order ....

Hyland M., Pitts A. [1987] "The Theory of Constructions: categorical semantics and topos theoretic models" Categories in Comp. Sci. and Logic, Boulder (AMS notes).

....even better, the Calculus of Constructions This approach hinges upon two things: ffl there is a realizability category C with the untyped s.n. terms as realisers, ffl this category C is a model of the type theory in question. We know from the works of Seely [See87] Pitts [Pit87] and Hyland [HP89] that the categorytheoretic interpretation of such sophisticated type theories as System F or the Calculus of Constructions places heavy demands on the structure of categories. For example, in the case of System F, we essentially need a cloven fibration E #p B (see e.g. B 85] such that: ....

.... toposes [Hyl82, Hyl88] see also [HRR90] Hyland shows that any pca U gives rise in a systematic way to a Kleene style realizability topos TOP s (U) which has more than sufficient completeness properties for interpreting at least the class of impredicative and dependent type theories [Pit87, HP89] To carry our programme through, the first step is to verify Assumption 0. 2 Strongly Normalising Untyped 3 Terms Our immediate task is the following: To construct a pca of strongly normalising untyped terms (or an appropriate quotient thereof) using the inherent application operator of the ....

J. M. E. Hyland and A. M. Pitts. The theory of constructions: Categorical semantics and topos-theoretic models. Contemporary Mathematics, 92:137--199, 1989.

.... however, been a surge of interest starting with the recent categorical description of models of the polymorphic lambda calculus [24] There is also a recent trend to using more general fibred categories in describing dependent type systems, see for example the recent papers of Hyland and Pitts [7], Jacobs [8] and Pavlovi c [16] Recall that constraint systems are given by a first order language interpreted over some structure and that they come equipped with a notion of entailment, conjunction and substitution. Our main task is to introduce existential quantification in terms of ....

J. M. E. Hyland and A. M. Pitts. The theory of constructions: Categorical semantics and topos-theoretic models. In Categories in Computer Science and Logic, pages 137--199. AMS, 1987. AMS Contemporory Mathematics Series 92.

....c pca U , there is a systematic way to construct a Kreiselstyle modified realizability topos TOP m (U) with elements of U as the realizers. Further, the topos TOP m (U) has remarkably good completeness properties which are (far more than) sufficient to provide a categorical semantics (see [HP89, Str92, Pit89] for a large class of higher type theories including, for example, all of Barendregt s Cube [Bar91] Two questions Putting the two results together, a generic s.n. argument may be assembled. It boils down to the following. To apply the generic argument to a type theory T , we ....

J. M. E. Hyland and A. M. Pitts. The theory of constructions: Categorical semantics and topos-theoretic models. Contemporary Mathematics, 92:137--199, 1989.

.... it) and showing that the Axiom of Choice is not derivable in CC (by constructing a model in which the type that represents the Axiom of Choice is empty) 1 Introduction In the literature there are many investigations on the semantics of polymorphic calculus with dependent types (see for example [12, 11, 10, 1, 5, 13]) Most of the existing models present a semantics for systems in which the inhabitants of the impredicative universe (types) are lifted to inhabitants of the predicative universe (kinds) see [16] Such systems are convenient to be modeled by locally Cartesian closed categories having small ....

....extensions of CC, such as inductive types and kinds (see [15] An interesting question is whether the whole model construction can be extended in a modular way to give semantics of richer systems than CC. We compare our notion of model with the following. Categorical Models(see for example [11]) We do not use the abstract machinery of category theory and instead present a simple, intuitively grounded notion of model for CC being a PTS. Standard Realizability Models (see [12, 13] The differences here are conceptual. As has been mentioned before, realizability models are a ....

J. M. E. Hyland and M. Pitts. The theory of constructions: Categorical semantics and topos-theoretic models. In Boulder, editor, AMS notes, 1987.

....y produces an n greater or equal to u(y) Here f n 2 N j n u(y) g is a well formed (subset) type in the context y : Y . For the technical development, we make use of B. Jacobs fibrational description of models of dependent type theory [19,21,22] which is related to the display map categories [37,17], categories with attributes [8,27] D categories [10] and comprehensive fibrations [29] See [19] for a comprehensive introduction. We make a point of describing the models in a so called split way, so as to avoid problems with interpreting dependent type theory. See, for example, ....

J.M.E. Hyland and A.M. Pitts. The theory of constructions: categorical semantics and topos theoretical models. In J.W. Gray and A. Scedrov, editors, Categories in Computer Science and Logic, volume 92 of Contemporary Mathematics, pages 137--199. AMS, Providence, RI, 1989, 1989.

....: Tp . B) 2 h) X not free in f (lX : Tp. fx) f : X : Tp . B) 5. 1 Definition: PN 2 is the deductive system defined above and lP 2 is the calculus of its proofs (as terms) The system PN 2 is a (revisited) fragment of Coquand Huet s Calculus of Constructions (see Hyland Pitts[1987]) 5.2 Remark: i) When ( I) is used w.r.t. to a non empty list G of assumptions, X is not free in uncancelled assumptions, by the same argument as in (4.1) ii) If there are no atomic types with free variables, lP 2 coincides with Girard Reynolds second order l calculus, since A B x:A.B . ....

....we will not discuss here the interpretation of dependent types, i.e. of PN 1 . To be precise, system F, as a pure theory, is equivalent to lP 2 . The ideas presented in this paper brought again to the limelight the realizability interpretation of impredicative type theory, as pointed out in in Hyland[1987]. A full categorical account of dependent types, including the theory of constructions, can be now found in Hyland Pitts[1987] see also Ehrhard[1988] and Robinson[1989] Call now M the interpretation of Tp. Then ( 2 I) roughly means that the interpretation of ( X:Tp.B) must be an element of ....

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Hyland M., Pitts A. [1987] "The Theory of Constructions: categorical semantics and topos theoretic models" Categories in C.S. and Logic, Boulder (AMS notes).

....is the first attempt to prove SN of a system which extends CC by an infinite type hierarchy, which also applies to GCC [Luo88b] The type checking problem for GCC is considered in [HaP88] because GCC does not have the property of type unicity, the resulted algorithm is rather complicated. In [HyP87] a general approach to categorical semantics of constructions like calculi is described, where an extension of constructions with Sigma types and unit type is presented with a motivation for discussing semantics. Ehr88] also gives a rather general framework of categorical semantics for dependent ....

M.Hyland and A.Pitts, `The Theory of Constructions: Categorical Semantics and Topos-theoretic Models', Categories in Computer Science and Logic, Boulder.

....5 Related Work The calculus of constructions (CC for short) is studied in [Coq85] CH88] CH85] etc. Its meta theory is developed in [Coq85] Coq86b] and [Pot87] There are several existing (independent) work on the semantic aspects of Constructions including the following. Hyland and Pitts [HPit87] developed a general approach to categorical semantics of Constructions like calculi, where an extension of CC with Sigma types and unit types is presented with the motivation of investigating semantics. Streicher [Str88] studied semantics of CC based on the notion of contextual category [Car86] ....

M. Hyland and A. Pitts, `The Theory of Constructions: Categorical Semantics and Topos-theoretic Models', Categories in Computer Science and Logic, Boulder.

....whose definition only differs in requiring the monoidal structure in the base is actually a product so that weakening and contraction can be interpreted. This notion of a cartesian handling of contexts is implicit in most of the work on categorical modelling of higher order typed calculi [Ehr88] [HP89]. Definition 2. Let B be a cartesian category with distinguished collection of objects T jBj. A cartesian context handling category is a functor E: B op Set T such that for each A 2 T there exists a natural isomorphism SubA : E( Gamma) A = Hom B ( Gamma; A) TermA We use Gamma; ....

J. Martin E. Hyland and Andrew M. Pitts. The theory of constructions: Categorical semantics and topos theoretic models. Contemporary Mathematics, 92:137--198, 1989.

....semantics of dependent products in Section 6.5. Several different, but interconnected, categorical structures have been proposed for interpreting the basic framework of dependent types by Seely [ 1984 ] Cartmell [ 1986 ] Taylor [ 1986 ] Ehrhard [ 1988 ] Streicher [ 1989, 1991 ] Hyland and Pitts [ 1989 ] Obtu lowicz [ 1989 ] Curien [ 1989 ] and Jacobs [ 1991 ] This reflects the fact that the categorical interpretation of dependent types is undoubtedly more complicated than the other varieties of categorical logic explained in this chapter. This is due to the structural complications ....

J. M. E. Hyland and A. M. Pitts. The theory of constructions: categorical semantics and topos-theoretic models. In 88 Andrew M. Pitts Gray and Scedrov [ 1989 ] , pages 137--199.

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Hyland, M. and Pitts A., [1989], "The Theory of Constructions: Categorical Semantics and ToposTheoretic Models".

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