A type theory for cartesian closed bicategories

Marcelo Fiore; Philip Saville

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arXiv · 1904.06538

A type theory for cartesian closed bicategories

Marcelo Fiore·Philip Saville

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Abstract

We construct an internal language for cartesian closed bicategories. Precisely, we introduce a type theory modelling the structure of a cartesian closed bicategory and show that its syntactic model satisfies an appropriate universal property, thereby lifting the Curry-Howard-Lambek correspondence to the bicategorical setting. Our approach is principled and practical. Weak substitution structure is constructed using a bicategorification of the notion of abstract clone from universal algebra, and the rules for products and exponentials are synthesised from semantic considerations. The result is a type theory that employs a novel combination of 2-dimensional type theory and explicit substitution, and directly generalises the Simply-Typed Lambda Calculus. This work is the first step in a programme aimed at proving coherence for cartesian closed bicategories.

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Marcelo Fiore, Philip Saville. 2019-04-13. A type theory for cartesian closed bicategories. https://arxiv.org/abs/1904.06538

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