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arXiv · 2410.19520

Synthetic 1-Categories in Directed Type Theory

Thorsten Altenkirch·Jacob Neumann
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Abstract

The field of directed type theory seeks to design type theories capable of reasoning synthetically about (higher) categories, by generalizing the symmetric identity types of Martin-Löf Type Theory to asymmetric hom-types. We articulate the directed type theory of the category model, with appropriate modalities for keeping track of variances and a powerful directed-J rule capable of proving results about arbitrary terms of hom-types; we put this rule to use in making several constructions in synthetic 1-category theory. Because this theory is expressed entirely in terms of generalized algebraic theories, we know automatically that this directed type theory admits a syntax model and is the first step towards directed higher observational type theory.

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Thorsten Altenkirch, Jacob Neumann. 2024-10-25. Synthetic 1-Categories in Directed Type Theory. https://doi.org/10.4230/lipics.types.2024.7

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