J.M.E. Hyland. The effective topos. In A.S. Troelstra and D. van Dalen (editors), The L.E.J. Brouwer Centenary Symposium. North Holland, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....dom. Synthetic Domain Theory Little is known about the exact links between the work of this thesis and similar ideas from synthetic domain theory. It is the case that complete Sigma spaces form a (constructive) model of the FIX= logic; for material relevant to synthetic domain theory see [Hyl82] and [Pho90] One could perform a routine inter translation of the systems to gain further incites into how they relate. 10.2 Final Conclusions We have presented three logical systems which can be used to interpret programming languages. These logics can be used to give meaning to both call by ....

J.M.E. Hyland. The effective topos. In A.S. Troelstra and D. van Dalen, editors, The L.E.J. Brouwer Centenary Symposium, Studies in Logic and the Foundation of Mathematics. North Holland, 1982.

....Menni, private communication) In general, the well complete regular Sigma posets form a full reflective category of sets [29] and provide yet another respectable notion of predomain. Well complete regular Sigma posets also arise very naturally in models. For example, in the effective topos, [7], the well complete regular Sigma posets are (equivalent to) the complete extensional PERs of [6] Even more strikingly, in Fiore and Rosolini s topos H [3, 4] the category of wellcomplete regular Sigma posets is equivalent to the familiar category of complete partial orders from classical ....

....(in a more restrictive setting) in [15] They justify the necessity, in general, of considering the somewhat clumsy notion of well completeness rather than the cleaner notion of completeness. In the brief Section 3 we revisit the best known model of SDT, that given by the Effective Topos ([7]) Our purpose here is to point out some unexpected ways in which the Effective Topos is less well behaved than one might expect. In particular, we show that the initial lift algebra is not an internal colimit of its standard chain of approximating iterates (it is trivially not an external 3 ....

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J.M.E. Hyland. The Effective Topos. In The L.E.J. Brouwer Centenary Symposium, A.S. Troelstra and D. van Dalen (eds), North-Holland 1982.

....generalizations [2, 4] all embed in Grothendieck toposes [3, 5] and hence, by [15, Ch. IV] in categories with class structure. Moreover, under mild conditions, Axiom N is satisfied. Also, by their very definition, realizability models [9, 20, 21, 22, 16, 17] embed in realizability toposes [10, 12], and hence in categories with class structure [15, Ch. IV] Again, Axiom N is satisfied. Thus, Theorem 1 gives an account of the construction of solutions to recursive domain equations that applies simultaneously to domain theoretic and to realizability models. As all nontrivial Grothendieck and ....

....determined by a divergence D A (see [17, Def. 4.1] such that Axiom 2 holds. Numerous examples are presented in [16, 17] Furthermore, by [17, Theorem 7.5] it follows that Axiom N holds. As is well known, there is a full embedding RT(A) of modest sets into the realizability topos over A [10, 12]. Assuming a strongly inaccessible cardinal, one can follow [15, and endow RT(A) with class structure. By constructing the initial ZF algebra V in RT(A) and then applying [31, Theorem 7] one extracts a full subcategory RT V (A) # RT(A) with class structure, in which V is a universal ....

J.M.E. Hyland. The effective topos. In L.E.J. Brouwer Centenary Symposium, pages 165--216. North-Holland, 1982.

....recursive, bijective pairing is such that: 0 (x) 0 for all x h0; 0i = 0 Then 0 2 U p ( for all . From this observation, Grayson, in an unpublished manuscript ( Gra81] gave a sketch how to build a modified realizability tripos and consequently a topos, in de style of Hyland s ([Hy82]) effective topos. In my thesis ( vO91] I filled in some details left blank by Grayson, and I observed that the Grayson topos is a sheaf subtopos of the effective topos built over Set . In the nineties, interest in modified realizability was revived. Streicher ( Str93] links the idea of ....

....to an interpretation of fully intensional type theory, via his category of modified assemblies. Hyland and Ong ( HyO93] see also Ong Ritter, OR94] give an account of modified realizability toposes based on conditional partial combinatory algebras. They develop some theory, analogous to [Hy82], and record the, at first sight surprising fact that there are two embeddings of Set into the modified realizability topos: one is the logical one, defined from the logic of the tripos; and the other is the direct image of the embedding of Set as : sheaves in the topos. Among others, this ....

J.M.E. Hyland, The Effective Topos, in: A.S. Troelstra & D. van Dalen (eds), The L.E.J. Brouwer Centenary Symposium, NorthHolland 1982

....can look at the structure of A recursive functions for a subset A ae IN. Computing these functions, one is allowed to consult an oracle which gives answers to the question x 2 A for any x; of course this begins to be interesting when A is not recursive. One has a topos Eff A and it is known ([Hy], P] that Eff A is a sheaf subtopos of Eff . Let RA be the analogon of R with respect to A partial recursive functions. One has the fibration (Eff A ) and it is likewise the exact completion of a left exact fibration ProjRA RA . Theorem 3.1 arises as a calculus of fractions ....

J.M.E. Hyland, The effective topos, in: Troelstra & Van Dalen (eds.), The L.E.J. Brouwer Centenary Symposium, Studies in Logic, Amsterdam 1982, pp. 165-216

....paper. According to me, there are three landmark publications in Realizability. These are: 1) Kleene s original 1945 paper, On the Interpretation of Intuitionistic Number Theory ( 51] 2) Troelstra s Metamathematical Investigations from 1973 ( 93] 3) Hyland s The Effective Topos from 1981 ([40]) Of these three, both 1) and 3) initiated a whole new strand of research. I have therefore decided that the material I wished to present, naturally divides into two periods, viz. 1940 1980 and 1980 2000. This is not to say that suddenly there were, after 1980, no more purely syntactical ....

....results. 2.1 The effective topos A completely new type of topoi was discovered around 1979 (apparently following some ideas of Scott; independently, there had been work of W.Powell along similar lines) by Martin Hyland, Peter Johnstone and Andy Pitts. The relevant publications are [44] 73] and [40]. It was well known, and amply demonstrated in Fourman and Scott s paper [25] that Boolean valued sets generalize to Heyting valued sets for a complete Heyting algebra. The completeness of the algebra is used for interpretation of the quantifiers. Now in [25] Fourman and Scott had dissected the ....

[Article contains additional citation context not shown here]

J.M.E. Hyland. The effective topos. In A.S. Troelstra and D. Van Dalen, editors, The L.E.J. Brouwer Centenary Symposium, pages 165--216. North Holland Publishing Company, 1982.

....we omit them, providing the reader only with the definitions and constructions involved. Lemma 2.5 Let RT[A ] refer to the realizability topos introduced in the previous section. Then Set (RT[A ] Proof. The geometric inclusion Gamma a r is defined exactly as for the Effective Topos (see [3]) Definition 2.6 An object (X; of RT[A ] is called canonically separated iff the equality on X satisfies [ x = x 0 ] 6= implies x = x 0 . An object is separated for the double negation topology iff it is isomorphic to some canonically separated object. Later on, we shall use the ....

J.M.E. Hyland. The effective topos. In A.S. Troelstra and D. Van Dalen, editors, The L.E.J. Brouwer Centenary Symposium, pages 165--216. North Holland Publishing Company, 1982.

.... (see, for example, 11] From the algebraic side, Partial Combinatory Algebras gave rise to the construction of elementary toposes as shown in [4] for every partial combinatory algebra A we have the realizability topos RT[A ] The best known of these toposes is Hyland s Effective Topos (see [3]) This paper is motivated by the question: what would be a good category for partial combinatory algebras (pca s) such that the construction of a realizability topos RT[A ] out of A becomes a functor with nice properties Of course, this depends on one s point of view as to which category these ....

J.M.E. Hyland. The effective topos. In A.S. Troelstra and D. Van Dalen, editors, The L.E.J. Brouwer Centenary Symposium, pages 165--216. North Holland Publishing Company, 1982.

....paper. According to me, there are three landmark publications in Realizability. These are: 1) Kleene s original 1945 paper, On the Interpretation of Intuitionistic Number Theory ( 47] 2) Troelstra s Metamathematical Investigations from 1973 ( 87] 3) Hyland s The Effective Topos from 1981 ([37]) Of these three, both 1) and 3) initiated a whole new strand of research. I have therefore decided that the material I wished to present, naturally divides into two periods, viz. 1940 1980 and 1980 2000. This is not to say that suddenly there were, after 1980, no more purely syntactical ....

....12 2.1 The effective topos A completely new type of topoi was discovered around 1979 (apparently following some ideas of Scott; independently, there had been work of W.Powell along similar lines) by Martin Hyland, Peter Johnstone and Andy Pitts. The relevant publications are [40] 68] and [37]. It was well known, and amply demonstrated in Fourman and Scott s paper [23] that Boolean valued sets generalize to Heyting valued sets for a complete Heyting algebra. The completeness of the algebra is used for interpretation of the quantifiers. Now in [23] Fourman and Scott had dissected the ....

[Article contains additional citation context not shown here]

J.M.E. Hyland. The effective topos. In A.S. Troelstra and D. Van Dalen, editors, The L.E.J. Brouwer Centenary Symposium, pages 165--216. North Holland Publishing Company, 1982.

.... widely used for the semantics of types over models of l calculus (see Scott (1976) Hindley (1983) It actually originated in Proof Theory, by work of Kreisel (see also Girard(1972) 10 Troelstra(1973) and is now widely used in the semantics of Intuitionistic Logic and higher order calculi (Hyland(1982 1987) Longo Moggi(1988) and many others) 3.1 Definition Let A = X, be an applicative structure. Define then: 1) The category PER A of partial equivalence relations given by: objects: R PER A iff R is an equivalence relation on a subset X R of X, i.e. X R = dom R = rangeR. morphisms: ....

Hyland M. (1982) "The effective Topos," in The Brouwer Symposium, (Troelstra, Van Dalen eds.) North-Holland.

.... calculi by Girard [9, 10] see also Tait [23] Various expositions and analyses of such proofs are given in Mitchell [15] Krivine [14] Huet [11] and Gallier [5, 6, 7, 8] among others. Another crucial concept is that of a partial equivalence relation, or PER. PER s were introduced by Hyland [12] and Mulry [17] PERs are a major tool in defining categories of domains in an effective setting (see Freyd, Mulry, Rosolini, and Scott [3] PERs also often show up as logical relations, and are called logical PERs (see Breazu Tannen and Coquand [2] In this note, we prove a general theorem for ....

J.M.E. Hyland. The effective topos. In A. S. Troelstra and D. Van Dalen, editors, L. E. J. Brouwer, Centenary Symposium, Studies in Logic. North-Holland, 1982.

....this form of comprehension. 1 Introduction In 1979 I was fortunate enough to attend some lectures in which Martin Hyland described, for the first time in public, how to use Kleene s notion of recursive realizability (Kleene 1945) to build what subsequently came to be known as the effective topos (Hyland 1982). Although motivated by applications in constructive analysis, this topos turned out to have some intriguing properties (Hyland 1988; Rosolini 1990) of use to the related fields of type theory and programming language semantics. For example, see (Phoa 1990) This is a preliminary version. The ....

Hyland, J. M. E. (1982). The effective topos. In A. S. Troelstra and D. van Dalen (Eds.), The L. E. J. Brouwer Centenary Symposium, pp. 165--216.

....to say the development of recursive versions of the main results in Algebra, Analysis, Topology, starting with Turing, the Russian Constructivist school, Nerode, Metakides, etc. see e.g. 6, 20] can be understood to be what takes place within a realizability interpretation (see e.g. remarks in [49]) although efforts to exploit this in a systematic way have as yet been limited (see e.g. Scedrov s article in [98] To sum up, IZF and related constructive systems, viewed within the realizability interpretations, provide a powerful formalism for synthesizing correct algorithms from proofs of ....

....is hPA ; Omega i with [ U = Omega V ] U V V U: We refer the reader to [48] for the details, and for a proof of a much stronger fact, namely that any category built from a tripos along the lines pursued above for realizability algebras is a topos. In said reference, and in [49, 89] it is shown that E ff A , and in particular E ff , are topostheoretic formulations of the realizability universe . Those principles we saw validated in the McCarty interpretation which make sense in local set theory, are true when suitably translated in the internal logic of E ff A . For ....

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Hyland, M. [1982], "The Effective Topos", in The L.E.J. Brouwer Centenary Symposium, North Holland.

....since (unlike position functors) lists are not closed under exponentials. Data categories are defined to provide a setting for the study of data functors. Examples of data categories include Sets, Pos (the category of bottomless, complete partial orders) 10] and Eff , the effective topos [4]. To the locos structure (lextensive categories with list objects) used to model shapely type constructors we must add cartesian closure for the higher order structure. The bulk of the paper is devoted to defining terms, establishing the above result, and showing that the data functors are closed ....

J.M.E. Hyland. The effective topos. In A.S. Troelstra and D. van Dalen (editors), The L.E.J. Brouwer Centenary Symposium. North Holland, 1982.

....profound generalisation of the idea that a theory is a category appears in the topos notion developed by Lawvere, Tierney, and others. In a sense, this notion captures the essence of set theory. It also has surprising relationships to algebraic geometry, computing science, and intuitionistic logic [36, 2, 42]. 9 Discussion The traditional view of foundations requires giving a system of axioms, preferably first order, that assert the existence of certain primitive objects with certain properties, and of certain primitive constructions on objects, such that all objects of interest can be constructed, ....

Martin Hyland. The effective topos. In A.S. Troelstra and van Dalen, editors, The Brouwer Symposium. North-Holland, 1982.

....I 2 B, the reindexing functor 3 P;I has a rightadjoint 5 P;I satisfying the Beck Chevalley condition where P;I : P 2 I I is the projection morphism. Happily, realizability categories satisfying the above requirements are available as appropriate sub structures of realizability 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 ....

.... instance, in the case of the effective topos Eff, the canonical realizability topos over the pca K 1 Kleene s first model of natural numbers, a sentence of Heyting Arithmetic is recursively realised if and only if it is internally true of the natural number object of the topos Eff (see [Hyl82] Hyland s pca to topos construction is actually an instance of a more general construction that yields a class of toposes as studied in Tripos Theory [HJP80] Formally, a tripos (which is an acronym for Topos Representing Indexed Pre Ordered Sets ) is a structure couched in the language of ....

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J. M. E. Hyland. The effective topos. In The L. E. J. Brouwer Centenary Symposium, pages 165--216. North-Holland, 1982.

....on the HEO (see [LM84] for references and a connecting theory of higher type functionals) The model had an essentially syntactic flavour. Via more recent ideas of Hyland and Moggi, though, it has acquired an independent interest as a small complete category, within Hyland s Effective Topos ( Hyl82,Hyl87] We will use this model here to discuss a fundamental aspect of parametricity: the so called uniformity of the definition of polymorphic terms. The objects of the category PER below are equivalence relations on subsets of the natural numbers or partial equivalence relations (p.e.r. s) ....

....below, or the intersection of p.e.r. s, is the object component of an internal product functor (see [AL91] also for references to the many authors who developed this matter) Notice that the embedding of the set PER as (PER, M ) into w Set is just the canonical embedding of Set into Eff, in [Hyl82] For our purposes, this embedding corresponds to the intuition that nothing can be said effectively when taking as inputs possibly infinite p.e.r. s; indeed, since M = w PER, realizers cannot help in distinguishing among p.e.r. s. Without going into internal adjunctions, the product in ....

M. Hyland, The effective Topos, in The Brouwer Symposium, (Troelstra, Van Dalen eds.) NorthHolland, 1982.

....These axiomatisations should be consistent with known categories of domains, and should also provide structural requirements for new semantic categories. History and state of the art The early work was by Robin Grayson, Martin Hyland, David McCarty, Phil Mulry and Giuseppe Rosolini (see [Hyl82, McC84, Mul81, Ros86] It focused on the effective topos Eff based on Kleene s realizability, and it was noted that effectively given Scott domains form a full sub category of Eff , and that there is an object Sigma of recursively enumerable truth values, which can be defined in the internal ....

.... this second form of description is that the standard PER model appears as a set theoretic model, provided that one understands set theoretic liberally enough allow models of intuitionistic set theory; the model of intuitionistic ZF that contains the PER model in this way is the effective topos [Hyl82, Ros90] From the point of view of intuitionistic set theory the result is models where the function type A B is interpreted by all set theoretic functions and 8 is an indexed product Q D2Type T (D) This is particularly startling, because Reynolds [Rey84] had earlier shown the impossibility ....

[Article contains additional citation context not shown here]

J. M. E. Hyland. The effective topos. In A. S. Troelstra and D. van Dalen, editors, The L. E. J. Brouwer Centenary Symposium, pages 165--216. North-Holland, 1982.

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J.M.E. Hyland. The effective topos. In A.S. Troelstra and D. van Dalen (editors), The L.E.J. Brouwer Centenary Symposium. North Holland, 1982.

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J.M.E. Hyland. The effective topos. In A.S. Troelstra and D. van Dalen, editors, The L.E.J. Brouwer Centenary Symposium. North Holland, 1982.

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J.M.E. Hyland. The effective topos. In A.S. Troelstra and D. van Dalen, editors, The L.E.J. Brouwer Centenary Symposium. North Holland, 1982.

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J.M.E. Hyland. The effective topos. In A.S. Troelstra and D. van Dalen, editors, The L.E.J. Brouwer Centenary Symposium. North Holland, 1982.

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J.M.E. Hyland. The effective topos. In The L.E.J. Brouwer Centenary Symposium, pages 165--216. North Holland, 1982.

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Hyland, J. M. E. The Effective Topos. in: The L. E. J. Brouwer Centenary Symposium, edited by A. S. Troelstra and D. van Dalen. Studies in Logic and the Foundations of Mathematics, vol. 110, North-Holland, Amsterdam, 1982, pp. 165-- 216.

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Hyland M. [1982] "The effective Topos," in The Brouwer Symposium, (Troelstra, Van Dalen eds.) North-Holland.

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