A map of sets is a process for getting from one set to another. We investigate the composition of maps (following one process by a second process), and find that the algebra of composition of maps resembles the algebra of multiplication of numbers, but its interpretation is much richer.

The goal of this book has been to show how the notion of composition of maps leads to the most natural account of the fundamental notions of mathematics, from multiplication, addition, and exponentiation, through the basic notions of logic and of connectivity.

Your further work with mathematics may apply it to physics, computer science or to other fields. In each of these, illuminating guides to the formulation and solution of problems often come from explicit recognition of structures occurring in commutative algebra, functional analysis, algebraic topology, etcetera. Clarifying unification of these branches has been developed during the last 60 years, using the categorical methods that you have begun to learn. To begin to deepen your knowledge of categories, here are four appendices which, although too brief for learning the subjects thoroughly, outline some important connections, in formulations which you will recognize in your subsequent encounters with mathematical topics.

Appendix I A general description of the geometry of figures in a space and the algebra of functions on a space, together with their basic functorial behavior.

Appendix II The description of Adjoint Functors and how they are exemplified in the categories of directed graphs and dynamical systems.

Appendix III A very brief history of the emergence of the theory of categories from within various mathematical subjects.

Appendix IV An annotated bibliography to guide you through elementary texts, monographs, and historical sources.

May you enjoy the fruits of your perseverance.

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.