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Metric spaces, generalized logic, and closed categories. (English) Zbl 0335.18006
The author gives an extensive introduction into the theory of \({\mathbf V}\)-categories for a closed category \({\mathbf V}\), more precisely a bicomplete symmetric monoidal closed category \({\mathbf V}\). In this article the author proceeds by first studying a concept in a typical example and then giving the abstract definition. – The metric spaces, mentioned in the title, are \({\mathbf R}\)-categories, where \({\mathbf R}\) is the closed category of nonnegative real numbers with the inverse order and the operations \(+\) and the truncated \(-\). \({\mathbf R}\)-functors are distance decreasing maps between metric spaces. Notions from logic appear in the context of partially ordered sets, which can be considered as 2-categories, since 2 is a closed category with the order and the operations of conjunction and implication. 2-functors are order preserving maps. \(\mathbf{Ab}\), the category of abelian groups, is closed and \(\mathbf{Ab}\)-categories are additive categories and, in particular, rings are \(\mathbf{Ab}\)-categories with one object. The notion of a bimodule generalizes the notion of an additive functor. – The author suggests to think of \(\mathbf V\)-categories as categories obeying a generalized logic determined by the closed category \(\mathbf V\). Thus the general notions for \(\mathbf V\)-categories like functor categories, bimodules, left and right Kan extensions, discrete fibrations and free categories reduce in the case \({\mathbf V}=2\) to the set of order ideals (a sort of power set), binary relations, existential and universal quantification along a map, the comprehension for characteristic functions and the transitive closure of a relation. Similarly, in the case \({\mathbf V}={\mathbf R}\), which should be thought of as a quantitative logic, the upper and lower integral of a real function can be viewed as Kan extensions. Moreover, the Cauchy-completeness of a metric space turns out to be the special case of a more general notion concerning the representability of bimodules.
Reviewer: H. Volger

MSC:
18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
18A15 Foundations, relations to logic and deductive systems
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