Given a subobject-structured category \(\mathcal X\), we construct a new category whose objects are the pairs (X, c) where X is an \(\mathcal X\)-object and c is an idempotent, monotonic and extensive endomap of the subobject lattice of X, and whose morphisms between objects are the closed maps between the corresponding subobject lattices. We give a sufficient condition on \(\mathcal X\) for the new category to be cartesian closed.

中文翻译：

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具有幂等闭合算子和闭合态射的类的笛卡尔闭合性

给定一个子对象结构化类别\（\ mathcal X \），我们构造了一个新类别，其对象是对（X，  c），其中X是\（\ mathcal X \）- object，而c是等幂，单调且X的子对象晶格的扩展内映射，其对象之间的态射是对应的子对象晶格之间的闭合映射。我们在\（\ mathcal X \）上给出足够的条件，以使新类别成为笛卡尔封闭的。