# Cheng’s Higher-dimensional category theory

Eugenia Cheng Higher-Dimensional Category Theory The architecture of mathematics (Thesis – University of Cambridge)
November 2000

## Introduction

The `high’ dimensions are not fully understood; various attempts have been made, but the relationship between them has
been unclear. Broadly speaking, my work has focused on the huge task of unifying the different theories.

## Chapter 1

### 1.1 Theory: What is mathematics?

Mathematics as a language has developed with a general aim of eliminating ambiguity. What has been sacrificed in pursuit of this ideal?

The most obvious sacrifice is that of scope. Rigour cannot be imposed upon every element of human consciousness. (Indeed, it may be precisely this impossibility that makes the human consciousness so endlessly rich.) In order to maintain rigour, we must be carefully precise about the issues we
are considering, and the context in which we are considering them.

A conceptual system is a system involving only ideas rather than physical phenomena. Physical systems pre-existing in the physical world around us already have properties which we can only observe and therefore not control. Scientific experiments seek to isolate parts of physical systems in order better to study their properties; a conceptual system might be seen as the purest form of such isolation. It is not only objects that are isolated, but characteristics of those objects.

Generally, we study such a system by defining the components we desire as our `building blocks’, together with any rules we require them to satisfy.

A small community may rely on the common sense of its inhabitants to preserve order. However, as the community grows it may become helpful or indeed necessary to organise the unspoken rules into a formal system of law. The system should reflect the `common sense’ behaviour of the inhabitants; the fact that it has been written down should aect their daily lives very little.

#### 1.2 Category Theory: The mathematics of mathematics

I have asserted that mathematics is the rigorous study of conceptual systems, and that category theory is the mathematics of mathematics. So category theory is the rigorous study of a conceptual system, where the system in question is mathematics itself.

[In] mathematics, it is not enough to know which objects we are considering; we must also specify the context in which we are relating these objects. Is a bicycle better than an egg? In the category `transport’, a bicycle is clearly better, but certainly not in the category `food’.

A category, then, is a collection of objects together with some ways of relating them to each other.

#### 1.3 Dimensions in Category Theory: Layers of complication

A category is a collection of objects together with some relationships between them. These relationships may also be regarded as objects and so
might also have relationships between them. These relationships might also have relationships between them, which might have relationships between them. Each of these levels of `relationships’ is what is called a dimension in category theory.

#### 1.4 Higher Dimensional Category Theory: Minimal rules for maximal expression

An n-category has

• objects: called 0-cells
• relationships between objects: called 1-cells
• relationships between relationships between objects: called 2-cells
• relationships between relationships between relationships between objects: called 3-cells
::: (all the above being building blocks)
• rules

The difficulty is that, just as relationships may have relationships, so rules may also satisfy rules. Rules for rules may also satisfy rules, and these themselves may satisfy more rules, and so on.

The difficulty is that as the number of dimensions increases, the complexity of the necessary rules increases with fearsome rapidity. … The thought of writing down the rules for a 5-category would make most category theorists shudder, let alone for a 10-category or a 4-million-category.
Clearly, some other way of approaching the theory is required.

## Chapter 2

### Completed research: The relationship between different approaches to higher-dimensional category theory.

There is an unmapped mountain. Various mountaineers
claim to have reached the summit; each has returned with
a map of his own route, and wondrous tales of the view
from the top. To map the whole mountain we must at
least see how the dierent routes relate to one another.
Did all the mountaineers really reach the top? In fact,
were they even climbing the same mountain?

## The future: up and along

As I progress towards the summit of the mountain I must
at each stage decide whether to proceed straight up the
face or to edge my way further around.

[The] so-called `coherence theorems’ … formalise the crucial assertion `this theory is sensible’.  …  It would … be of great value to be able to understand the coherence issue in a general n-dimensional setting, without having to consider each dimension separately.

There remains the important idea that the collection of n-categories should itself form an (n+1)-category … the need
for a generalisation to n is indisputable. It is therefore of utmost importance that this matter be resolved in order for any theory of n-categories to be at all satisfactory.