个体集和强个体集的范畴性质  

Categorical properties of individual sets and strong individual sets

作　　者：李生刚[1] 杨文华[1] 伏文清[1] 

机构地区：[1]陕西师范大学数学与信息科学学院,陕西西安710062

出　　处：《陕西师范大学学报（自然科学版）》2013年第1期1-4,14,共5页Journal of Shaanxi Normal University：Natural Science Edition

基　　金：国家自然科学基金资助项目(11071151);陕西省自然科学基金资助项目(2010JM1005)

摘　　要：研究了个体集和强个体集的范畴性质.利用范畴论方法证明了个体集范畴、强个体集范畴与集合范畴在许多方面是相似的.例如,具有任一给定基数的个体集和强个体集是存在的;个体集和强个体集对于子集、幂运算封闭;非空个体集范畴和非空强个体集范畴都是完备的monoidaltopoi.构造了超结构函子V和超幂函子HF并得到:(1)对任意非空强个体集X和Y,g:X→Y是单射(resp.,满射)当且仅当V(g)是单射(resp.,满射);(2)对任意集X和Y,g:X→Y是单射(resp.,满射)当且仅当HF(g)是单射(resp.,满射).The category of individual sets and the category of strong individual sets are proved to be much similar to the category of sets by using method of category theory. For example, for each cardinality a, there exists an individual set （resp. , a strong individual set） X. such that ｜Xa｜ =a,where ｜Xa｜ is the cardinality of Xa; individual sets and strong individual sets are closed under the operations of subsets and powers; both the category of nonempty individual sets and the category of nonempty strong individual sets are complete monoidal topoi. Superstructure functor V and ultrapower functor H~ are constructed and the following conclusions are obtained. （1） For any nonempty strong individual sets X and Y, a map g：X--~Y is an injection （resp. , a surjection） if and only if V（g） is an injection （resp. , a surjection）; （2） For any sets X and Y,a map g：X→Y is an injection （resp. , a surjection） if and only if HF（g） is an injection （resp. , a surjection）.

关 键 词：个体集 强个体集 范畴 完备范畴 MONOIDAL范畴 TOPOS 超结构函子 超幂函子 

分 类 号：O141.41[理学—数学] O154.1[理学—基础数学]