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An internal language for autonomous categories

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Abstract

We present an internal language for symmetric monoidal closed (autonomous) categories analogous to the typed lambda calculus as an internal language for cartesian closed categories. The language we propose is the term assignment to the multiplicative fragment of Intuitionistic Linear Logic, which possesses exactly the right structure for an autonomous theory. We prove that this language is an internal language and show as an application the coherence theorem of Kelly and Mac Lane, which becomes straightforward to state and prove. Finally, we extend the language with the natural numbers and show that this corresponds to a weak Natural Numbers Object in an autonomous category.

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Authors and Affiliations

  1. Department of Computing, Imperial College of Science, Technology and Medicine, SW7 2BZ, London, England

    Ian Mackie, Leopoldo Román & Samson Abramsky

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  1. Ian Mackie
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  2. Leopoldo Román
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  3. Samson Abramsky
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Mackie, I., Román, L. & Abramsky, S. An internal language for autonomous categories. Appl Categor Struct 1, 311–343 (1993). https://doi.org/10.1007/BF00873993

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