归纳数据类型的范畴论方法

Category Theoretical Method of Inductive Data Types

归纳数据类型是类型论研究的重要分支,传统的数理逻辑或代数方法侧重于描述归纳数据类型的有限语法构造,在语义性质与归纳规则的分析与设计方面存在一定的不足。基于范畴论的方法,在集合范畴的框架内给出谓词的形式化定义,分析谓词范畴与代数范畴的构成与性质,并探讨集合范畴上自函子到谓词范畴上自函子的提升,最后利用伴随函子及其伴随性质深入分析了归纳数据类型具有普适意义的归纳规则。

Inductive data types are important research branch of type theory, and traditional methods including mathe- matical logic and algebra focus on describing finite syntax construction for inductive data types, resulting in some defi- ciencies for analyzing and designing semantics properties and inductive rules. This paper provided formal definition of predicate in the framework of set category by category theoretical methods, analyzed the construction and properties of predicate category and algebra category,and probed lifting of endofunctors from set category to predicate category, and at last deeply researched universal inductive rules of inductive data types by adjoint functor and its adjoint properties.