J.M.E. Hyland. Filter spaces and continuous functionals. Annals of Mathematical Logic, 16, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....section we will consider the hereditarily total objects in the standard domain interpretations of the types, and compare this hierarchy with our main hierarchy. We will assume familiarity with the theory of algebraic domains, e.g. as introduced in Stoltenberg Hansen, Lindstr Sm and Griffor [14] Convention In this paper we will normally let a domain D be an alge braic domain, or a Scott Ershov domain in the sense of [14] that satisfies coherence, i.e. a subset X of D is bounded in D if and only if any two point subset of X is bounded in D. 2 2 The types Definition 1 We define the ....

....hierarchy with our main hierarchy. We will assume familiarity with the theory of algebraic domains, e.g. as introduced in Stoltenberg Hansen, Lindstr Sm and Griffor [14] Convention In this paper we will normally let a domain D be an alge braic domain, or a Scott Ershov domain in the sense of [14] that satisfies coherence, i.e. a subset X of D is bounded in D if and only if any two point subset of X is bounded in D. 2 2 The types Definition 1 We define the type terms inductively as follows: 1. The constants R, N, B and are type terms. 2. If o and are type terms, then (o x ) and ....

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Hyland, J.M.E. Filter Spaces and Continuous Functionals, Annals of Mathematical Logic 16, (1979) 101-143.

....function and product types. Thus these hierarchies admit the definition of a general notion of continuous functionals; we call these continuous functionals of transfinite type and consider this as a natural extension of the notion of continuous functionals of finite type as studied in e.g. [4, 8, 10]. The purpose of the present paper is to provide a topological characterisation of these functionals. For the continuous functionals of finite type a characterisation using limit spaces has been given by Scarpellini [15] We extend the typed hierarchy of limit spaces to the transfinite case, and ....

Hyland, J.M.E. Filter Spaces and Continuous Functionals, Annals of Mathematical Logic 16, 101-143, (1979).

.... is not cartesian closed ( 44] and references therein) and regular epis are not stable under pullback [25] recall that an epi map is called regular if it is the coequalizer of some pair of maps) These imperfections motivated research in order to find better categories of spaces (see for example [24, 70, 38, 46, 59] and more recently [102, 8, 21, 77, 78] There is an obvious forgetful functor : Top # Set that assigns to each topological space its underlying set. This functor has both a left adjoint # and a right adjoint #, both of which are full and faithful. The functor # : Set # Top assigns to ....

J. M. E. Hyland. Filter spaces and continuous functionals. Annals of mathematical logic, 16:101--143, 1979.

....recursion theory and in order theoretic characterizations of extensionality for total objects [4, 17] It is important to understand how di erent models of computation relate. Indeed, a number of results demonstrate that the Kleene Kreisel functionals arise in various computational models [7, 10, 15, 3, 13], which is good evidence that this class of functionals is an important and robust model of higher type computation. We proved one such result in [2] where we related domains with totality to equilogical spaces, introduced by Dana Scott [2] the so called dense and codense totalities on domains ....

J.M.E. Hyland. Filter spaces and continuous functionals. Annals of Mathematical Logic, 16, 1979.

....the category of topological spaces to a full subcategory that is cartesian closed. Some well known examples are: Steenrod s category of compactly generated Hausdor # spaces (Mac Lane 1971) the category, Seq, of sequential spaces (which contains many computationally important non Hausdor# spaces) (Hyland 1979ii) or the even larger category of quotients of exponentiable spaces considered in (Day 1972) However, the received wisdom about such categories is that their function spaces are topologically hard to understand. It is much quoted that the exponential N N N can never be firstcountable ....

....1979ii) or the even larger category of quotients of exponentiable spaces considered in (Day 1972) However, the received wisdom about such categories is that their function spaces are topologically hard to understand. It is much quoted that the exponential N N N can never be firstcountable (Hyland 1979ii) whereas an ideal approach from a computational viewpoint would allow e#ectivity issues to be addressed, and the stricter requirement of secondcountability is often claimed to be necessary for such (see e.g. Smyth 1992) A second alternative is to expand the category Top by adding new ....

[Article contains additional citation context not shown here]

J. M. E. Hyland. Filter spaces and continuous functionals. Annals of Mathematical Logic, 16:101--143, 1979.

....recursion theory and in order theoretic characterizations of extensionality for total objects [4, 17] It is important to understand how different models of computation relate. Indeed, a number of results demonstrate that the Kleene Kreisel functionals arise in various computational models [7, 10, 15, 3, 13], which is good evidence that this class of functionals is an important and robust model of higher type computation. We proved one such result in [2] where we related domains with totality to equilogical spaces, introduced by Dana Scott [2] the so called dense and codense totalities on domains ....

J.M.E. Hyland. Filter spaces and continuous functionals. Annals of Mathematical Logic, 16, 1979.

....(0) s n 1 (0) and are the identity otherwise. To overcome these problems, we have decided to replace Cpo with the category Rep of replete objects in some model (E ; of SDT (see [Tay91] This note establishes some basic facts about replete objects in the category Fil of lter spaces (see [Hyl79]) namely: SFP Rep, where means full subcategory (see Theorem 2.4) a characterisation of the regular subobjects (in Rep) of an SFP (see Theorem 2.7) the following axiom is valid in Fil: m:X Y regular mono implies (m) regular epi, provided Y is a topological space (see ....

.... a characterisation of the regular subobjects (in Rep) of an SFP (see Theorem 2.7) the following axiom is valid in Fil: m:X Y regular mono implies (m) regular epi, provided Y is a topological space (see Theorem 2. 9) 1 Filter spaces: basic de nitions facts Basic de nitions (see [Hyl79]) F is a lter over X i is a non empty collections of nonempty subsets of X s.t. U V X;U 2 F V 2 F and U; V 2 F (U V ) 2 F F is a lter base over X i is a non empty collections of nonempty subsets of X s.t. U; V 2 F 9W 2 F:W (U V ) if F is a lter base, then [F ] ....

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J.M.E. Hyland. Filter spaces and continuous functionals. Annals of Mathematical Logic, 16, 1979.

....lattices [2] see also Section 3.2) Assemblies are an immediate generalisation of modest sets; the cartesian closed category ASSM of assemblies over algebraic lattices contains the whole of TOP, not just TOP 0 . Yet another such extension was already known in the seventies: filter spaces [6, 5, 1] also form a cartesian closed extension of TOP. In fact, there are three such extensions as each of the three cited texts comes with its own notion of filter space; the resulting three categories are contained in each other: FIL a FIL b FIL c TOP. All three are cartesian closed, but ....

....Scott continuous. We also have f [A] f A] for A X , hence f [ and f [x] fx] for x in X . The assignment f 7 f is functorial. 2.2 Filter Spaces in the Broadest Sense There are several notions of filter spaces in the literature. The most liberal one is the following [6]: A filter space is a set X together with a relation # between PhiX and X such that [x] # x holds for all x in X , and A # x and A B implies B # x. A function f : X Y between two filter spaces is continuous if A # x implies f A # fx. The category of filter spaces and continuous ....

[Article contains additional citation context not shown here]

J. M. E. Hyland. Filter spaces and continuous functionals. Annals of Mathematical Logic, 16:101--143, 1979.

....cut down the category of topological spaces to a full subcategory that is cartesian closed. Some well known examples are: Steenrod s category of compactly generated Hausdorff spaces [19] the category, Seq, of sequential spaces (which contains many computationally important non Hausdorff spaces) [12]; or the even larger category of quotients of exponentiable spaces considered in [6] However, the received wisdom about such categories is that their function spaces are topologically hard to understand. It is much quoted that the exponential N N N can never be first countable [12] whereas an ....

....spaces) 12] or the even larger category of quotients of exponentiable spaces considered in [6] However, the received wisdom about such categories is that their function spaces are topologically hard to understand. It is much quoted that the exponential N N N can never be first countable [12], whereas an ideal approach from a computational viewpoint would allow effectivity issues to be addressed, and the stricter requirement of second countability is often claimed to be necessary for such (see e.g. Smyth [27] A second alternative is to expand the category Top by adding new objects ....

[Article contains additional citation context not shown here]

J.M.E. Hyland. Filter spaces and continuous functionals. Annals of Mathematical Logic, 16:101--143, 1979.

....= s n 1 (0) and are the identity otherwise. To overcome these problems, we have decided to replace Cpo with the category Rep of replete objects in some model (E ; Sigma) of SDT (see [Tay91] This note establishes some basic facts about replete objects in the category Fil of filter spaces (see [Hyl79]) namely: ffl SFP ae Rep, where ae means full subcategory (see Theorem 2.4) ffl a characterisation of the regular subobjects (in Rep) of an SFP (see Theorem 2.7) ffl the following axiom is valid in Fil: m:X Y regular mono implies Sigma(m) regular epi, provided Y is a topological ....

....of the regular subobjects (in Rep) of an SFP (see Theorem 2.7) ffl the following axiom is valid in Fil: m:X Y regular mono implies Sigma(m) regular epi, provided Y is a topological space (see Theorem 2. 9) 1 Filter spaces: basic definitions facts Basic definitions (see [Hyl79]) ffl F is a filter over X iff is a non empty collections of nonempty subsets of X s.t. U V X;U 2 F oe V 2 F and U; V 2 F oe (U V ) 2 F ffl F is a filter base over X iff is a non empty collections of nonempty subsets of X s.t. U; V 2 F oe 9W 2 F:W (U V ) if F is a filter base, then ....

[Article contains additional citation context not shown here]

J.M.E. Hyland. Filter spaces and continuous functionals. Annals of Mathematical Logic, 16, 1979.

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J.M.E. Hyland. Filter spaces and continuous functionals. Annals of Mathematical Logic, 16, 1979.

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J.M.E. Hyland, Filter Spaces and Continuous Functionals, Annals of Mathematical Logic, 16 (1979), 101--143.

....Moldestad [10] establishing the conection between Platek s notion of computability and Kleene s notion. Kleene computability over Ct( c) c is weaker than being recursive. This was first proved by Tait [15] who showed that the fan functional is 2 not computable, and further explored in Hyland [5] and in Normann [12, 13] Kleene computabiliW is also known to be weaker than PCF definabiliW. Berger [2] showed that the fan functional is indeed PCF definable. By a straightforward application of the recursion theorem, we can show that the functional F introduced by Gandy, see Hyland [5] is ....

....Hyland [5] and in Normann [12, 13] Kleene computabiliW is also known to be weaker than PCF definabiliW. Berger [2] showed that the fan functional is indeed PCF definable. By a straightforward application of the recursion theorem, we can show that the functional F introduced by Gandy, see Hyland [5], is again PCF computable. Berger [2] conjectured that any recursire functional in Ct(k) is PCF definable. The main result of this paper is that this conjecture is true. It makes a great difference if we work with Ct(k) or with the hereditarily total functionals in P(k) Plotkin [15] showed that ....

Hyland. J.M.E. Filterspaces and continuous functionals, Annals of Mathematical Logic 16 (1979) 101 - 143.

....same. Berger [2] observed that the fan functional, shown by Tait [17] not to be computable, is indeed computable if the computations take place in the P hierarchy. By a straitforward application of the recursion theorem, we can show that the functional F introduced by Gandy and showed by Hyland [5] not to be computable in the fan func tional, is computable over the P hierarchy. Berger [2] conjectured that any recursive functional is computable in the P hierarchy. The main result of this paper is that this conjecture is true. The recursion theory of P(k) c has also atracted the interest ....

Hyland. J.M.E. Filterspaces and continuous functionals, Annals of Mathematical Logic 16 (1979) 101 - 143.

....then t is a classi er in C I , the partial map classi ers exist and are preserved by reindexing, and q is closed under the partial map classi er. When S = j where j is a topology, then S f 1 is the quasi topos of j sheaves. An interesting example is the quasi topos of lter spaces (see [Hyl79]) where topological spaces embed as a full re ective subcategory (which is not closed under exponentiation) When S is (the lter space corresponding to) Sierpinski s space, SFP domains form a full sub CCC of S f 1 . Example 1.13 Given an extensive cartesian closed category D with small ....

J.M.E. Hyland. Filter spaces and continuous functionals. Annals of Mathematical Logic, 16, 1979.

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J.M.E. Hyland. Filter spaces and continuous functionals. Annals of Mathematical Logic, 16:101-- 143, 1979.

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