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Relevant categories and partial functions. (English) Zbl 1164.03002
The authors introduce the notion of relevant category as a symmetric monoidal closed category with a diagonal natural transformation satisfying some coherence conditions. The denomination ‘relevant’ comes from its connection with relevant logic. Namely, as intuitionistic propositional logic may be identified with the bicartesian closed category freely generated by a set of propositional letters, so the positive fragment of intuitionistic relevant logic may be identified with a free relevant category. Although every Cartesian closed category is a relevant category, it is shown that the category of sets with partial functions, which is isomorphic to the category of pointed sets, is a category that is relevant, but not Cartesian closed.

MSC:
03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
03G30 Categorical logic, topoi
18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
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