论文标题
内部邻里结构III：子对象的有限总和
Internal Neighbourhood Structures III: Finite Sum of Subobjects
论文作者
论文摘要
具有有限索引的有限完整类别的内部preneighbourhood空间的概念以及适当的$（\ MathSf {e}，\ Mathsf {M}）$系统，因此对于每个对象$ x $，$ \ mathsf {M} $ x $ $ x $的$ x $ semitiate是一个完整的lattice。封闭操作员的概念，封闭的形态及其接近的盟友在\ cite {2021-clos}中进行了调查。本文提供了三重态$（\ mathbb {a}，\ mathsf {e}，\ mathsf {m}）$（带有$ \ mathbb {a} $ lextention等于$ $ \ mathsf {m mathsf {m} $ - subobjects of Bignite Intunite Insunite Indife Intimite Sums。还提供了在有限总和下封闭的一组封闭嵌入（封闭形态）的等效条件。如果在有限总和下封闭了可允许的亚对象（分别是封闭的嵌入）的晶格，则显示有限总和的可允许亚对象（分别是封闭的嵌入）的联接半层次被证明是组件连接半层次的双iProduct。最后，每当有限总和下封闭封闭形态的集合时，就会显示出来，在有限的总和下，适当（分别，分开的）形态的一组也会封闭。这导致了在有限总和下关闭的紧凑型（分别是豪斯多夫）的全部子类别的等效条件。
The notion of an internal preneighbourhood space on a finitely complete category with finite coproducts and a proper $(\mathsf{E}, \mathsf{M})$ system such that for each object $X$ the set of $\mathsf{M}$-subobjects of $X$ is a complete lattice was initiated in \cite{2020}. The notion of a closure operator, closed morphism and its near allies investigated in \cite{2021-clos}. The present paper provides structural conditions on the triplet $(\mathbb{A}, \mathsf{E}, \mathsf{M})$ (with $\mathbb{A}$ lextensive) equivalent to the set of $\mathsf{M}$-subobjects of an object closed under finite sums. Equivalent conditions for the set of closed embeddings (closed morphisms) closed under finite sums is also provided. In case when lattices of admissible subobjects (respectively, closed embeddings) are closed under finite sums, the join semilattice of admissible subobjects (respectively, closed embeddings) of a finite sum is shown to be a biproduct of the component join semilattices. Finally, it is shown whenever the set of closed morphisms is closed under finite sums, the set of proper (respectively, separated) morphisms are also closed under finite sums. This leads to equivalent conditions for the full subcategory of compact (respectively, Hausdorff) preneighbourhood spaces to be closed under finite sums.
