范畴内部算子中的开态射

Open morphisms in categorical interior operators

给出范畴内部算子的一些结果,通过一般拓扑学中连续开映射的等价刻画,定义了范畴内部算子中的开态射,并研究了它们的性质.设Ω是一个范畴,ω是Ω上的一类单态射使得(ε,ω)是一个恰当的保持的分解系统.IN(Ω,ω),CL(Ω,ω)和NO(Ω,ω)分别记为范畴Ω相对应ω的范畴内部算子、范畴闭包算子和范畴邻域算子的全体.当满足一定条件和适当的序关系给IN(Ω,ω),CL(Ω,ω)和NO(Ω,ω),可以证明它们彼此是完备类之间的同构.

Some results of categorical interior operators were first given here; then open morphisms in categorical interior operators were defined and their properties studied by using the equivalent characterization of continuous open mappings in general topology. Let D be a category and a fixed class ω of Ω-monomorphisms such that （ε, ω） is a proper stable factorization system. IN（Ω,ω）, CL（Ω,ω） and NO（Ω,ω） denote the set of all categorical closure operators on Ω with respect to ω, the set of all categorical neighborhood operators on D with respect to co and the set of all categorical interior operators on Ω with respect to ω. When it satisfies some conditions and appropriate order relations can be defined on IN（Ω,ω）, CL（Ω,ω） and NO（Ω,ω）, it can be proven they are isomorphisms between complete classes.