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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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\textit{F. W. Lawvere}, Rend. Semin. Mat. Fis. Milano 43, 135--166 (1974; Zbl 0335.18006)

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### References:

[1] | MacLane S.,Categories for the Working Mathematician Springer 1972. · Zbl 0705.18001 |

[2] | Eilenberg S. & Kelly G. M.,Closed Categories, in Proceedings of La Jolla Conference on Categorical Algebra. Springer 1966. · Zbl 0192.10604 |

[3] | Benabou Jean,Les Distributeurs, Raport n^o 33, janvier 1973, Seminaires de Mathématiques Pure, Institut de Mathématiques, Université Catholique de Louvain (multigraphed). |

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