J. Lambek and P.J. Scott: Introduction to higher order categorical logic. Cambridge University Press, Cambridge 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....principle every replete object is a Sigma cpo. Section 6 explains how on top of both variants the main results of classical domain theory can be proved quite easily. 2 Logical Preliminaries of Synthetic Domain Theory We formulate the axioms of Synthetic Domain Theory in a topos logic (cf. Lambek, Scott 86] i.e. a higher order intuitionistic logic with axiom of unique choice, arithmetic and subtype formation. Let N denote the natural numbers and Omega the type of propositions. Recall that therefore we have extensionality on functions, i.e. 8f; g:A B: 8a:A:fa = ga) f = g and on propositions ....

J. Lambek and P.J. Scott: Introduction to higher order categorical logic. Cambridge University Press, Cambridge 1986.

....The framework for our discussion is a category whose objects are the sets of closed terms of a closed type. 4. 1 Definitions and basic properties Recall that given a typed l calculus language and a l theory T, a category Cl(T) is determined by taking as objects of Cl(T) the (closed) types of T [LS 86] MS 89] As for morphisms, choose first one variable for each type and define the morphisms from A to B to be equivalence classes of typing judgments x:A # t:B, where x is the chosen variable of type A, and the equivalence relation is given by the equality judgments x:A # tt :B of T. We will ....

J.Lambek, P.J.Scott: Introduction to higher order categorical logic, Cambridge University Press, 1986.

....product types as nite products and function types as 22 power objects; terms are interpreted as morphisms. abstraction then just corresponds to applying the co universal property of the power objects. In fact, calculi are even in a certain sense equivalent to cartesian closed categories [28]. The correspondence works roughly as follows: One can construct a cartesian closed category from a typed calculus by taking types as objects and terms as morphisms; conversely, a cartesian closed category gives rise to a calculus which has the objects as types and the morphisms as function ....

J. Lambek and P. J. Scott: Introduction to Higher Order Categorical Logic, Cambridge University Press, 1986.

....The framework for our discussion is a category whose objects are the sets of closed terms of a closed type. 4. 1 Definitions and basic properties Recall that given a typed l calculus language and a l theory T, a category Cl(T) is determined by taking as objects of Cl(T) the (closed) types of T [LS 86] MS 89] As for morphisms, choose first one variable for each type and define the morphisms from A to B to be equivalence classes of typing judgments x:A # t:B, where x is the chosen variable of type A, and the equivalence relation is given by the equality judgments x:A # tt :B of T. We will ....

J.Lambek, P.J.Scott: Introduction to higher order categorical logic, Cambridge University Press, 1986.

....calculational motivation of the well known finitary term rewriting (cf. 2] and [4] It would still have its use, however, in demonstrating nonequivalence. For example, the theory of categories with certain infinite products may be axiomatized by an infinite set of infinitary equations. cf. [5] for the finite case. One way of showing terms of this theory to be nonequivalent would be to interpret them as distinct arrows in some category of sets. As an application of the results of this note, we shall derive another method: the theory in question may be so oriented as to constitute a ....

J. Lambek & P.J. Scott: Introduction to higher order categorical logic. Cambridge 1986.

.... products if there is a linear functor x: X X X, x = and x = and linear transformations : Id X x X : X X, r) and i : x X 2 i : X X X, i = p i ; b i ) i = 0; 1) Furthermore, these must satisfy the standard equations for cartesian products (as in [LS86] for example) 0 x 1 = id x : x x : X X X ; i = id : id id : X X (i = 0; 1) Note that given our de nition of composition of linear transformations, these are equivalent to the usual equations for products (in the monoidal coordinate) and coproducts (in the comonoidal coordinate) ....

J. Lambek and P.J. Scott Introduction to Higher-Order Categorical Logic. Cambridge Studies in Advanced Mathematics 7, Cambridge University Press (1986).

....U , and F = F 0 ; U . These ideas are closely related to the treatment of names in the action calculi, as analyzed by Pavlovi c [P96] There he shows that to add an indeterminate to a symmetric monoidal category in a manner which is functionally complete (in the sense of Lambek and Scott [LS86]) is equivalent to moving to the Kleisli category induced by a commutative comonoid. He then uses this to explain the treatment of names in the static part of the action calculus. Thus, a commutative monoidal structural action provides a means to add functionally complete indeterminates at all ....

Lambek, J. and P.J. Scott Introduction to Higher-Order Categorical Logic. Cambridge studies in advanced mathematics 7, Cambridge University Press, 1986.

....introductory books on the subject exist, e.g. Arbib Manes 75,Rydeheard Burstall 88] The mathematically inclined reader may prefer [MacLane 71] but this can hardly be recommended as an introduction for the average computer scientist. For the higher order constructs the main reference is [Lambek Scott 86] Unlike many other branches of mathematics, category theory is basically an organizational framework, and may therefore seem very abstract. While a number of non trivial theorems can be stated for the pure theory, much of the work involving categories consists of applications of the ....

....B not containing x, such that f x ffi hx ffi 3;idi = f . Dually, for a variable y : T 0, there exists a unique morphism f y : A T B, such that [2 ffi y; id] ffi f y = f . Proof The detailed verifications are rather verbose, but simple in principle. In fact, many of the CCC results from [Lambek Scott 86] can be carried over directly, verifying that only the weaker forms of the axioms are used. Since the SCL axioms are symmetrical, the results dualize immediately to continuation abstractions. First, we define a number of auxiliary morphisms, expressing associativity, commutativity and ....

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Joachim Lambek and P.J. Scott: Introduction to Higher Order Categorical Logic. Cambridge studies in advanced mathematics, Vol. 7, Cambridge University Press (1986).

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J. Lambek and P.J. Scott: Introduction to Higher--Order Categorical Logic, Cambridge Studies in Advanced Math. no. 7 (Cambridge University Press, 1983).

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J. Lambek and P. Scott Introduction to Higher-Order Categorical Logic, CUP, 1986.

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J. Lambek and P. J. Scott: Introduction to higher order categorical logic, Cambridge U. P. (1986).

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Lambek, J. and P.J. Scott Introduction to Higher-Order Categorical Logic. Cambridge studies in advanced mathematics 7, Cambridge University Press (1986).

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