G. Rosolini, Continuity and Eectiveness in Topoi, Ph. D. Thesis, University of Oxford (1986).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....characterized in terms of the Heyting algebra H associated with the span f; In what follows, P (H) def E H op E ; Y : H P (H) will denote the topos of presheaves associated with a poset H in E , with its Yoneda functor Y. A description of P (H) E as a continuous bration [24] can be given as follows. Let C denote the underlying small category of a subcanonical site for E over S. Denote by H the following category. The objects of H are pairs (c; x) where c is an object of C and c x H is a morphism in E . A morphism (c; x) f (d; y) in H is a morphism c f d ....

G. Rosolini, Continuity and eectiveness in topoi, Ph. D. Thesis, University of Oxford (1986).

....essential to this paper. The subobject classi er was originally de ned by Bill Lawvere in 1969, and the basic theory of elementary toposes was developed in collaboration with Myles Tierney during the following year [Law71, Law00] Giuseppe Rosolini generalised the de nition to classes of supports [Ros86] and developed a theory of partial maps [RR88] but the Frobenius law (Propositions 3.11, 8.2 and 10.13) is also required for relational algebra. 2.1. Definition. A class M of morphisms (written , of any category C, such that (a) all isomorphisms are in M, b) all M maps are mono, so i 2 M ....

....be many classes M of supports, each class possibly being classi ed by some object M . 2.13. Remark. Many of the ideas in this paper evolved from synthetic domain theory, a model of which is a topos (with a classi er for all monos) that also has a classi er for recursively enumerable subsets [Ros86, Pho90a, Pho90b, Hyl91, Tay91, FR97, BR98] In this case, is a subsemilattice of Such models exist wherein the full subcategory of replete objects satis es the monadicity property discussed in this paper for , in addition to that for the whole category for [RT97] A distilled account of ....

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Giuseppe Rosolini. Continuity and Eectiveness in Topoi. D. Phil. thesis, University of Oxford, 1986.

....At the end we get a formal system, the computational lambda calculus ( c calculus for short) for proving equivalence of programs, which is sound and complete w.r.t. the categorical semantics of computations. 1 The methodology outlined above is inspired by [13] 2 , and it is followed in [11, 8] to obtain the p calculus. The view that category theory comes, logically, before the calculus led us to consider a categorical semantics of computations rst, rather than to modify directly the rules of conversion to get a correct calculus. A type theoretic approach to partial ....

....language (like ML) cannot be interpreted by the exponential B A (as done in a ccc) is fairly obvious; in fact the application of a functional procedure to an argument requires some computation to be performed before producing a result. By analogy with partial cartesian closed categories (see [8, 11]) we will interpret functional types by exponentials of the form (TB) A . De nition 2.6 A c model over a category C with nite products is a strong monad (T ; t) together with a T exponential for every pair hA; Bi of objects in C, i.e. a pair h(T B) A ; eval A;TB : T B) A A) ....

G. Rosolini. Continuity and Eectiveness in Topoi. PhD thesis, University of Oxford, 1986.

....that can be found only after developing a semantics based on mathematical structures rather than term models, but it does not give clear criteria to single out the general principles among the properties satis ed by the model. The approach adopted in this paper generalises the one followed in [Ros86, Mog86] to obtain the p calculus, i.e. the calculus for reasoning about partial computations (or equivalently, about partial functions) In fact, the p calculus (like the calculus) amounts to a particular c theory . A type theoretic approach to partial functions and computations is attempted ....

....approach to partial functions and computations is attempted in [CS87, CS88] by introducing a new type constructor A, whose intuitive meaning is the set of computations of type A. However, Constable and Smith do not adequately capture the general axioms for (partial) computations as we (and [Ros86]) do, since they lack a general notion of model and rely only on domain and recursion theoretic intuition. 2 A categorical semantics of computations The basic idea behind the semantics of programs described below is that a program denotes a morphism from A (the object of values of type A) to TB ....

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G. Rosolini. Continuity and Eectiveness in Topoi. PhD thesis, University of Oxford, 1986.

....good extension of EN , the recursive topos R, such that EN is embedded in R and the embedding preserves limits ( nite colimits) and function spaces. In Section 4 we introduce a topos similar to R and give a topos theoretic characterization of GEN , following quite closely Chapter 6 of [Ros86] From this characterization and general facts in topos theory (see [Hyl82] Section 5 and [Ros86] Chapter 6) one can easily derive the properties of GEN , that in Section 2 are proved by elementary means. In Section 3 we compare a type structure in GEN with other ones, introduced in ....

....preserves limits ( nite colimits) and function spaces. In Section 4 we introduce a topos similar to R and give a topos theoretic characterization of GEN , following quite closely Chapter 6 of [Ros86] From this characterization and general facts in topos theory (see [Hyl82] Section 5 and [Ros86] Chapter 6) one can easily derive the properties of GEN , that in Section 2 are proved by elementary means. In Section 3 we compare a type structure in GEN with other ones, introduced in connection with Recursion Theory in higher types. From this comparison it turns out that morphisms ....

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G. Rosolini. Continuity and Eectiveness in Topoi. PhD thesis, Oxford, 1986.

....in order to interpret richer languages, in particular the calculus. 4. We show that w.l.o.g. one may consider only (monads over) toposes, and we exploit this fact to establish conservative extension results. The methodology outlined above is inspired by [Sco80] 1 , and it is followed in [Ros86, Mog86] to obtain the p calculus. The view that category theory comes, logically, before the calculus led us to consider a categorical semantics of computations rst, rather than to modify directly the rules of conversion to get a correct calculus. Related work The operational approach to nd ....

.... The categorical semantic of computations presented in this paper has been strongly in uenced by the reformulation of Denotational Semantics based on the category of cpos, possibly without bottom, and partial continuous functions (see [Plo85] and the work on categories of partial morphisms in [Ros86, Mog86]. Our work generalises the categorical account of partiality to other notions of computations, indeed partial cartesian closed categories turn out to be a special case of c models (see De nition 3.9) A type theoretic approach to partial functions and computations is proposed in [CS87, CS88] ....

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G. Rosolini. Continuity and Eectiveness in Topoi. PhD thesis, University of Oxford, 1986.

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G. Rosolini, Continuity and Eectiveness in Topoi, Ph. D. Thesis, University of Oxford (1986).

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