A type theory for cartesian closed bicategories (Extended Abstract)

Abstract

Index Terms-typed lambda calculus, higher category theory, Curry-Howard-Lambek correspondence, cartesian closed bicategoriesAbstract-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.

Coherence and normalisation-by-evaluation for bicategorical cartesian closed structure

“…This settles a conjecture put forward by Ouaknine [59]. In the context of our [32] it establishes that, modulo the equational theory of cartesian closed bicategories, there is at most one rewrite between any two terms in the type theory Λ ×,→ ps for cartesian closed bicategories on a set of base types.…”

“…Our second proof, by contrast, is novel. Using the 'internal language' Λ ×,→ ps for cartesian closed bicategories presented in [32], we reduce the problem of coherence to a normalisation problem for Λ ×,→ ps , which we solve using semantic methods. This 'denotational semantics' approach to coherence, which we outline in Section 4, is guided by two principles.…”

“…First, Theorem 1.1 drastically reduces the difficulty of constructing structure in an arbitrary cartesian closed bicategory. This will have applications in the development of a theory of opetopic structures [26] in the cartesian closed bicategories of generalised species [27] and cartesian distributors [29], as well as in the study of the equational theory of rewriting in the STLC [32,33].…”

Relative Full Completeness for Bicategorical Cartesian Closed Structure

“…The result may also be expressed type-theoretically. For instance, in terms of the type theories of [20], the type theory Λ ×,→ ps for cartesian closed bicategories is a conservative extension of the type theory Λ × ps for finite-product bicategories. It follows that, modulo the equational theory of bicategorical products and exponentials, any rewrite between STPC-terms constructed using the βη-rewrites for both products and exponentials may be equally presented as constructed from just the βη-rewrites for products (see [21,55]).…”

“…On the other hand, the importance of glueing in the categorical setting suggests that its bicategorical counterpart will find a range of applications. A case in point, which has already been developed, is the proof of a 2-dimensional normalisation property for the type theory Λ ×,→ ps for cartesian closed bicategories of [20] that entails a corresponding bicategorical coherence theorem [21,55]. There are also a variety of syntactic constructions in programming languages and type theory that naturally come with a 2-dimensional semantics (see e.g.…”