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

Foundations

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?

Chapter 3

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.

My Comments

This contains some important insights and an important ‘call to arms’.

My own view is that if, for some given definition of ‘category’, the collection of (n+1)-categories is not necessarily an (n+1) category then the notion of category ought to be broadened to some kind of quasi-category which is so closed.

As presented, category theory seems to be aiming at improving conceptual systems based on ‘building blocks’. Thus at any given time mainstream physics often has very basic ‘atoms’, ‘particles’ or other building blocks, and so category theory seems positioned to help improve such theories. But, as with physics, our experience is often that what may seem like the most basic building blocks, or level-0 categories, often turn out to need to need re-casting as categories of categories. In general, we cannot be sure of any maximum level that we may need to consider, and this is clearly problematic, as we have no concept of consistency that spans multiple levels. For example, between sub-atomic particles to social phenomena there is more than one intermediate level, so we have no satisfactory concept of what it would mean to apply nuclear physics to our needs, except by making some gross assumptions. (Certainly, I have never found one.)

More prosaically, I am not even sure that there are not ‘categories all the way down’. That is, we may need a notion of quasi-category that recognizes that there is no ‘well-founded’ level-0 category, so that reality is not ‘well-grounded’. But we might be able to work around this limitation of n-category theory by recognizing that a particular n-category theory is only a conceptual system, and its level-0 might need underpinning. In practice, it might be useful to decorate category theory by performing a ‘pre-mortem’. That is, one imagines situations which might force an underpinning of level-0, and notes aspects that one might look at first and aspects that one would wish to preserve. For example, in physics one might represent experiments as morphism and note that any change needs to preserve their actual results, even if it changes their interpretation.

On a more basic level, it is perhaps worth noting that category theory assumes that morphisms are associative . I have often found that applications of mathematics often assume that this is so when perhaps it isn’t, and a lack of associativity is sometimes an indication that the theory needs amending. Hopefully associativity is something that an extended ‘ω-quasi-category theory’ would satisfy, else we would be really in trouble. But it would still be something that needed to be checked against experience, not just assumed.

Dave Marsay

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