作为谱空间的子对象空间
Spaces of subobjects as spectral spaces
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中文总结 AI 辅助
本文研究合适范畴中固定对象的子对象空间的自然拓扑，比较λ-生成对象与λ-紧元素概念，引入范畴论Zariski拓扑并得到谱性判据。
中文摘要 AI 辅助
我们研究合适范畴中固定对象的子对象空间上的自然拓扑，目的是确定这些空间何时为谱空间。我们方法的核心步骤（本身也具有独立意义）是比较范畴论中的λ-生成对象概念与子对象格中λ-紧元素的序论概念，证明在一些自然且温和的假设下这两个概念一致。在有限情形下，这使我们能仅用序论术语描述有限生成子对象，并在每个子对象格区间内构造一个典范代数核；我们针对完全格抽象研究该构造，并通过泛性质对其进行刻画。随后我们引入子对象空间上的范畴论Zariski拓扑，将其与Scott拓扑关联起来，得到整个子对象空间及其代数核的谱性判据。
英文摘要
We study natural topologies on spaces of subobjects of a fixed object in a suitable category, with the aim of determining when these spaces are spectral. A central step in our approach, which is also of independent interest, is the comparison between the categorical notion of a $λ$-generated object and the order-theoretic notion of a $λ$-compact element in a lattice of subobjects. We prove that these notions coincide under some natural and mild assumptions . In the finitary case, this allows us to describe the finitely generated subobjects purely in order-theoretic terms and to construct, inside each interval of a subobject lattice, a canonical algebraic core. We study this construction abstractly for complete lattices and characterize it by a universal property. We then introduce the categorical Zariski topology on spaces of subobjects, relate it to the Scott topology, and obtain spectrality criteria for the whole subobject space and for its algebraic core.
