A subunit in a monoidal category is a subobject of the monoidal unit for which a canonical morphism is invertible. They correspond to open subsets of a base topological space in categories such as those of sheaves or Hilbert modules. We show that under mild conditions subunits endow any monoidal category with a kind of topological intuition: there are well-behaved notions of restriction, localisation, and support, even though the subunits in general only form a semilattice. We develop universal constructions completing any monoidal category to one whose subunits universally form a lattice, preframe, or frame.

中文翻译：

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张量拓扑

幺半群范畴中的子单元是正则态射可逆的幺半群单元的子对象。它们对应于类别中的基本拓扑空间的开子集，例如滑轮或希尔伯特模块的类别。我们表明，在温和条件下，亚基赋予任何幺半群范畴一种拓扑直觉：尽管亚基通常只形成一个半格，但存在良好的限制、定位和支持概念。我们开发了将任何幺半群类别完善到其子单元普遍形成格子、预框架或框架的通用结构。