拓扑中的Lawvere完备性
Lawvere completeness in Topology
摘要：
自1973年以来，Lawvere的（Cauchy-）完备充实范畴的概念对于度量空间是有意义的：它精确地捕获了Cauchy-完全度量空间。本文引入了$(\mathbb{T},\mathsf{V})$ -范畴的Lawvere完备性的概念，并证明了它对拓扑空间和拟一致空间具有有趣的意义：前者是弱清醒性，后者是Cauchy完备性。此外，我们表明，$\mathsf{V}$有一个典型的$(\mathbb{T},\mathsf{V})$类别结构，发挥了关键作用：它是Lawvere完全在合理的条件下设置，允许我们定义一个Yoneda嵌入领域的$(\mathbb{T},\mathsf{V})$类别。
英文摘要：
It is known since 1973 that Lawvere's notion of (Cauchy-)complete enriched category is meaningful for metric spaces: it captures exactly Cauchy-complete metric spaces. In this paper we introduce the corresponding notion of Lawvere completeness for $(\mathbb{T},\mathsf{V})$-categories and show that it has an interesting meaning for topological spaces and quasi-uniform spaces: for the former ones means weak sobriety while for the latter means Cauchy completeness. Further, we show that $\mathsf{V}$ has a canonical $(\mathbb{T},\mathsf{V})$-category structure which plays a key role: it is Lawvere-complete under reasonable conditions on the setting; permits us to define a Yoneda embedding in the realm of $(\mathbb{T},\mathsf{V})$-categories.
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