Using the category of metric spaces as a template, we develop a metric analogue of the categorical semantics of classical/intuitionistic logic, and show that the natural notion of predicate in this “continuous semantics” is equivalent to the a priori separate notion of predicate in continuous logic, a logic which is independently well-studied by model theorists and which finds various applications. We show this equivalence by exhibiting the real interval $[0,1]$ in the category of metric spaces as a “continuous subobject classifier” giving a correspondence not only between the two notions of predicate, but also between the natural notion of quantification in the continuous semantics and the existing notion of quantification in continuous logic.Along the way, we formulate what it means for a given category to behave like the category of metric spaces, and afterwards show that any such category supports the aforementioned continuous semantics. As an application, we show that categories of presheaves of metric spaces are examples of such, and in fact even possess continuous subobject classifiers.

中文翻译：

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度量空间和连续逻辑的分类语义

使用度量空间的范畴作为模板，我们开发了经典/直觉逻辑的范畴语义的度量类似物，并表明谓词在这个“连续语义”中，等价于先验分离的谓词概念连续逻辑，一种由模型理论家独立充分研究并找到各种应用的逻辑。我们通过展示真实区间来展示这种等价性$[0,1]$在度量空间的范畴中，作为“连续子对象分类器”，不仅给出了两个谓词概念之间的对应关系，而且还给出了连续语义中的自然量化概念与连续逻辑中现有的量化概念之间的对应关系。 ，我们制定了给定类别表现得像度量空间类别的含义，然后证明任何此类类别都支持上述连续语义。作为一个应用，我们展示了度量空间的预滑层类别就是这样的例子，实际上甚至拥有连续的子对象分类器。