Quantales、广义premetrics和自由语言环境
Quantales, generalised premetrics and free locales
摘要：
预度量和可预度量空间已经被研究了很长时间，它们的拓扑相互关系也很好地理解。考虑预度量空间范畴${\bf Pre}$和$ε$ - $δ$连续函数作为态射。三角不等式的缺失意味着忠实函子${\bf Pre} \to {\bf Top}$ --一个预度量空间被发送到它生成的拓扑空间--不是满的。此外，从${\bf Pre}$中的对象生成的拓扑空间的序列性质表明，这个函子也不是对象上的满射。从Flagg和Weiss的工作，我们说明了一个扩展${\bf Pre}\hookrightarrow {\bf P} $连同一个忠实和满射的对象左伴随函子${\bf P} \to {\bf Top}$作为一个扩展的${\bf Pre} \to {\bf Top}$。我们表明，这是一个最佳的情况下，${\bf Pre} \to {\bf Top}$只保留余产品。${\bf P}$中的对象是度量类对象值的值分配格的限制和colimits，我们显示所产生的离散集上的自由locales。
英文摘要：
Premetrics and premetrisable spaces have been long studied and their topological interrelationships are well-understood. Consider the category ${\bf Pre}$ of premetric spaces and $ε$-$δ$ continuous functions as morphisms. The absence of the triangle inequality implies that the faithful functor ${\bf Pre} \to {\bf Top}$ - where a premetric space is sent to the topological space it generates - is not full. Moreover, the sequential nature of topological spaces generated from objects in ${\bf Pre}$ indicates that this functor is not surjective on objects either. Developed from work by Flagg and Weiss, we illustrate an extension ${\bf Pre}\hookrightarrow {\bf P} $ together with a faithful and surjective on objects left adjoint functor ${\bf P} \to {\bf Top}$ as an extension of ${\bf Pre} \to {\bf Top}$. We show this represents an optimal scenario given that ${\bf Pre} \to {\bf Top}$ preserves coproducts only. The objects in ${\bf P}$ are metric-like objects valued on value distributive lattices whose limits and colimits we show to be generated by free locales on discrete sets.
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