Connectedness versus discreteness

Besides map spaces and the truth space, another construction that is characterized by a ‘higher universal mapping property’ objectifies the counting of connected components. Reflexive graphs and discrete dynamical systems, though very different categories, support this ‘same’ construction. For example, we say that dots d and d′ in a reflexive graph are connected if for some n ≥ 0 there are

The graph as a whole is connected if it has at least one dot and any two dots in it are connected; it is noteworthy that to prove a given graph to be connected may involve arbitrarily long chains of elementary connections ai, even though the structural operators s, t, i are finite in number (Sessions 13 to 15).

By contrast, this aspect of steps without limit does not arise in the same way for dynamical systems, even though the dynamical systems themselves involve infinitely many structural operators αn, effecting evolution of a system for n units of time. We say that the states x, y are connected if there are n, m such that αnx = αmy, and that the system is connected if it has at least one state and every two states are connected.

We investigate the analogy: If composition of maps is like multiplication of numbers, what is like division of numbers? The answers shed light on a great variety of problems, including (in Session 10) ‘continuous’ problems.

Introduction

Our goal in this book is to explore the consequences of a new and fundamental insight about the nature of mathematics which has led to better methods for understanding and using mathematical concepts. While the insight and methods are simple, they are not as familiar as they should be; they will require some effort to master, but you will be rewarded with a clarity of understanding that will be helpful in unravelling the mathematical aspect of any subject matter.

The basic notion which underlies all the others is that of a category, a ‘mathematical universe’. There are many categories, each appropriate to a particular subject matter, and there are ways to pass from one category to another. We will begin with an informal introduction to the notion and with some examples. The ingredients will be objects, maps, and composition of maps, as we will see.

While this idea, that mathematics involves different categories and their relationships, has been implicit for centuries, it was not until 1945 that Eilenberg and MacLane gave explicit definitions of the basic notions in their ground-breaking paper ‘A general theory of natural equivalences’, synthesizing many decades of analysis of the workings of mathematics and the relationships of its parts.

Galileo and the flight of a bird

Let's begin with Galileo, four centuries ago, puzzling over the problem of motion. He wished to understand the precise motion of a thrown rock, or of a water jet from a fountain.

The unification of mathematics is an important strategy for learning, developing, and using mathematics. This unification proceeds from much detailed work that is punctuated by occasional qualitative leaps of summation. The 1945 publication by Samuel Eilenberg and Saunders Mac Lane of their theory of categories, functors, and natural transformations, was such a qualitative leap. It was also an indispensable prerequisite for a further leap, the 1958 publication by Daniel Kan of the theory of adjoint functors. The application of algebra to geometry had forced Eilenberg and Mac Lane to create their general theory; geometric methods developed by Alexander Grothendieck on the basis of that general theory were used 50 years later in the Andrew Wiles proof of Fermat's Last Theorem and in many other parts of algebra.

In the 1940s, the application which had given rise to the Eilenberg and Mac Lane summation, namely the study of qualitative forms of space in algebraic topology, began to be worked out by Eilenberg & Steenrod and others, and this development still continues in this century.

In the 1950s Mac Lane categorically characterized linear algebra; Yoneda showed that maps in any category can be represented as natural transformations; and Grothendieck made profound applications to the continuously-variable linear algebra which arises in complex analysis.

We find there is a single definition of multiplication of objects, and a single definition of addition of objects, in all categories. The relations between addition and multiplication are found to be surprisingly different in various categories.