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In domain theory, the basic idea is to find constructions from partial orders that can model lambda calculus, by creating a well-behaved cartesian closed category.
在域理论中,基本想法是通过建立良好行为的笛卡儿闭范畴,从可建模lambda演算的偏序中找到构造。
A category is called cartesian closed if it has finite direct products and a morphism defined on a finite product can always be represented by a morphism defined on just one of the factors.
A topos is a certain type of cartesian closed category in which all of mathematics can be formulated (just like classically all of mathematics is formulated in the category of sets).
A topos is a certain type of cartesian closed category in which all of mathematics can be formulated (just like classically all of mathematics is formulated in the category of sets).
For instance, the simply typed lambda calculus corresponds to the internal language of cartesian closed categories.
例如,简单类型λ演算 对应于笛卡儿闭范畴 (CCC)的内部语言。
In category theory, a category is considered Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors.
C is a cartesian closed category (or a topos) if and only if D is cartesian closed (or a topos).
C 是一个笛卡儿闭范畴(或一个拓扑斯)当且仅当 D 是笛卡儿闭(或拓扑斯)。
The first suggestion (1962) to remedy this situation was to restrict oneself to the full subcategory of compactly generated Hausdorff spaces, which is in fact cartesian closed.
Typed lambda calculi are closely related to mathematical logic and proof theory via the Curry-Howard isomorphism and they can be considered as the internal language of classes of categories, e.g. the simply typed lambda calculus is the language of Cartesian closed categories (CCCs).
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