Mathematics can be considered as the process of engaging with, understanding, and exploiting patterns. The strengths of the human mind are not perfectly fitted to the abstract problems which arise in this process. However, it is often possible to recruit our better intuitive and conceptual structures by rephrasing the problems in suitable terms. Much of mathematics therefore consists of setting up and refining amplified metaphors. For example, a great deal of mathematics is phrased in terms of geometry, using terms like shape and dimension. My research concerns game theory, which similarly makes use of analogies involving competitive interaction. More precisely, I am dealing with two-player complete-information games. The phrase `complete information' specifies that these are games which do not rely on chance (like snakes and ladders) or concealed data (like battleship). Instead, as in chess or noughts and crosses, either player always has enough information to completely describe both the current state of the game and how it will be modified by any legal move.
Though the theory describing such games is so elegant that it is worthy of study for its own sake, it can also be used to throw light on other areas of mathematics, principally logic and the theory of computation. Mathematical logic is the abstract study of mathematical reasoning itself, and so it is concerned with (simplified models of) the language with which such reasoning is usually expressed. Wittgenstein introduced the metaphor `Languages are games': one strand of mathematical logic has deepened and extended this metaphor for the simplified models of language studied by logicians. To each statement in such a language, we can associate a game, played by two players (the Challenger and the Defender) with the property that play in this game looks like the kind of discussion which might arise when determining whether the statement is true. Taking a very simple example, play in the game corresponding to the statement `Everybody has a mother' might look like this:
Challenger: What about [person A]?The Defender is declared to have won if person B is the mother of person A. We might then say that the statement is true if (in principle) the Defender has a winning strategy for this simple game, and false if the Challenger has a winning strategy. Not only does this help us to explore the meaning of words like `everybody,' it also allows us to recruit our conceptual understanding of games to help us think more fruitfully about mathematical language.
Defender: His/her mother is [person B].
The second major use of game theory is in the theory of computation. Here the subroutines of a program are thought of as players in a game, the structure of which guides the flow of the computation. On a larger scale, the protocols by means of which programs interact with one another may be thought of as basic games. This metaphor may be made precise and extended to show how standard concepts from computer science (parallel processing, variable binding, etc.) correspond to concepts involving games; seeing the concepts from this new perspective allows new ways of thinking about them.
Game theory is just one of the diverse species of mathematics, and can appear as different from ideas such as the convoluted geometry used to model the behaviour of spacetime on quantum scales as a poodle is from a blue whale. However, just as the poodle and the whale have strikingly similar skeletal structures, so there is a common structure under the surface, not just of these two areas, but of an incredibly diverse menagerie of mathematical fields. The study of this structure is category theory, the comparative anatomy of mathematics. [As is often true in mathematics, the word `category' has a specific technical meaning in this context, which has little to do with the normal usage. The best policy is to imagine that a completely new word is being introduced, and to disregard any standard meanings or connotations the word may have for you.]
One major benefit of category theory is that it provides a general linguistic framework within which many areas of mathematics may be discussed. This aids communication between mathematicians working in different fields, and assists in the recognition and development of connections between those fields. Indeed, in mathematics, simply expressing ideas in the right language can be a powerful aid to thought, suggesting new perspectives and approaches and reducing complex problems and definitions to simple ones. Category theory also allows concepts and techniques to be more easily transferred from one area to another, and guides the construction of the analogies and amplified metaphors on which mathematics thrives. For particularly thorny problems, category theory can help to identify simpler contexts which serve as guinea pigs for potential solutions; seeing what works in these simpler contexts can be useful in deciding what approach to take to the original problem. Finally, many of the structures made explicit in category theory are extraordinary for their simplicity and beauty, and these qualities are highly valued within the mathematical community.
Category theory was applied to the theory of games with great success in the latter half of the 20th century. During this time, a clear intuitive picture of how categorical structures might emerge in the theory of games was developed. However, when attempts have been made to make this intuitive picture more concrete, the details have proved to be rather fiddly. Several concrete explanations have emerged, each with its own peculiarities and with none evidently more natural than the others. At present, these explanations are held together only by a loose weave of suggestive analogies.
Some recent developments in category theory may change that. The widespread use of geometric intuitions in mathematics means that often the idea of dimension is key. We naturally assign dimensions to physical objects, so that a line on a piece of paper is just 1-dimensional, the surface of the paper itself is 2-dimensional, the space in which the paper sits is 3-dimensional and so on. In just the same way, mathematicians naturally assign dimensions to many of the abstract objects that we study, and this helps us to visualise them. When category theory first emerged, about halfway through the 20th century, the approach was entirely 1-dimensional. It quickly became clear that there were higher-dimensional analogues of the structures being studied, and that these higher-dimensional structures would allow a more expressive linguistic framework. To balance this expressivity, however, these structures are also more delicate and more care is needed to understand them. It is only in the last decade that the necessary techniques for handling these structures have been established, and there is still plenty of room for progress in this exciting area.
The categorical approaches to the understanding of games have so far been almost exclusively 1-dimensional. In my research, I have discovered a hidden extra dimension of structure underlying the intuitive picture of the relationship between categories and games. There is a natural way to make this precise using the fresh language of higher dimensional category theory. When this is applied to the disparate existing concretisations of the intuitive picture, the 2-dimensional structures I obtain show their unity more clearly than their 1-dimensional shadows. This gives a new way of looking at the existing constructions and the links between them, as well as allowing me to make concrete some constructions which had so far only been discussed on an intuitive level. This work has also provided a context for the development of some basic tools in higher-dimensional category theory.
In my PhD thesis, I hope to give a clear explanation of the higher dimensional construction, including both an introduction to an elegant general framework for the application of category theory to game theory and a detailed exposition of a simple new example. Beginning in parallel with this thesis, but continuing into the next couple of years of my research, I hope to publish a series of papers making use of this general framework to provide new perspectives on the existing constructions in this field, and on the connections between them. I hope to explain how the new framework makes possible a proof of a conjecture of Imre Leader. Finally, I hope to explore some of the problematic issues from the theory of games in the context of my new simple example, and so to indicate potential resolutions of these problems.
