Cartesian closed bicategories: type theory and coherence

TL;DR

This thesis extends the Curry--Howard--Lambek correspondence to bicategories, establishing a coherence theorem for cartesian closed bicategories through a novel type theory and normalization techniques.

Contribution

It introduces biclones for algebraic synthesis, proves a new coherence theorem without rewriting, and generalizes key categorical results to bicategories.

Findings

Proves at most one 2-cell between parallel 1-cells in free cartesian closed bicategories.

Develops a type theory satisfying local coherence and normalization.

Generalizes category-theoretic results to the bicategorical setting.

Abstract

In this thesis I lift the Curry--Howard--Lambek correspondence between the simply-typed lambda calculus and cartesian closed categories to the bicategorical setting, then use the resulting type theory to prove a coherence result for cartesian closed bicategories. Cartesian closed bicategories---2-categories `up to isomorphism' equipped with similarly weak products and exponentials---arise in logic, categorical algebra, and game semantics. I show that there is at most one 2-cell between any parallel pair of 1-cells in the free cartesian closed bicategory on a set and hence---in terms of the difficulty of calculating---bring the data of cartesian closed bicategories down to the familiar level of cartesian closed categories. In fact, I prove this result in two ways. The first argument is closely related to Power's coherence theorem for bicategories with flexible bilimits. For the second, which is the central preoccupation of this thesis, the proof strategy has two parts: the construction of a type theory, and the proof that it satisfies a form of normalisation I call "local coherence". I synthesise the type theory from algebraic principles using a novel generalisation of the (multisorted) abstract clones of universal algebra, called "biclones". Using a bicategorical treatment of M. Fiore's categorical analysis of normalisation-by-evaluation, I then prove a normalisation result which entails the coherence theorem for cartesian closed bicategories. In contrast to previous coherence results for bicategories, the argument does not rely on the theory of rewriting or strictify using the Yoneda embedding. Along the way I prove bicategorical generalisations of a series of well-established category-theoretic results, present a notion of glueing of bicategories, and bicategorify the folklore result providing sufficient conditions for a glueing category to be cartesian closed.