Synthetic 1-Categories in Directed Type Theory

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.