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.

The problem of finding map objects in a given category is complicated by the fact that often the map object we are looking for does not exist. This difficulty occurs many times in mathematics: we have a problem and we do not even know whether it has a solution. In such cases it is often helpful to pretend that the problem does have a solution, and proceed to calculate anyway! We need an account of

which we will then apply to the problem of determining map objects in the category of sets and in the category of graphs.

We imagine that we have already found the solution to a given problem, and try to deduce consequences from its existence. We ask ourselves: What does this solution imply? In this way we are often able to deduce enough properties of that solution to discover the real way to the solution or to prove that a solution is impossible.

To apply this method there are two parts, both of which are important. The first is to find out as much as possible about the solution one seeks under the assumption that a solution does exist. Usually one proves first a conclusion of the following type: If a solution exists it must be a certain thing. But the thing found may not be a solution. The second part consists in verifying that, indeed, this thing really is a solution to the problem.

We use maps to express extra ‘structure’ on sets, leading to graphs, dynamical systems, and other examples of ‘types of structure.’ We then investigate ‘structure-preserving’ maps.

Since its first introduction over 60 years ago, the concept of category has been increasingly employed in all branches of mathematics, especially in studies where the relationship between different branches is of importance. The categorical ideas arose originally from the study of a relationship between geometry and algebra; the fundamental simplicity of these ideas soon made possible their broader application.

The categorical concepts are latent in elementary mathematics; making them more explicit helps us to go beyond elementary algebra into more advanced mathematical sciences. Before the appearance of the first edition of this book, their simplicity was accessible only through graduate-level textbooks, because the available examples involved topics such as modules and topological spaces.

Our solution to that dilemma was to develop from the basics the concepts of directed graph and of discrete dynamical system, which are mathematical structures of wide importance that are nevertheless accessible to any interested high-school student. As the book progresses, the relationships between those structures exemplify the elementary ideas of category. Rather remarkably, even some detailed features of graphs and of discrete dynamical systems turn out to be shared by other categories that are more continuous, e.g. those whose maps are described by partial differential equations.

Many readers of the first edition have expressed their wish for more detailed indication of the links between the elementary categorical material and more advanced applications.

Balls, spheres, fixed points, and retractions

The Dutch mathematician L.E.J. Brouwer (1881–1966) proved some remarkable theorems about ‘continuous’ maps between familiar objects: circle, disk, solid ball, etc. The setting for these was the ‘category of topological spaces and continuous maps.’ For our purposes it is unnecessary to have any precise description of this category; we will instead eventually list certain facts which we will call ‘axioms’ and deduce conclusions from these axioms. Naturally, the axioms will not be selected at random, but will reflect our experience with ‘cohesive sets’ (sets in which it makes sense to speak of closeness of points) and ‘continuous maps.’ (Roughly, a map f is continuous if f(p) doesn't instantaneously jump from one position to a far away position as we gradually move p. We met this concept in discussing Galileo's idea of a continuous motion of a particle, i.e. a continuous map from an interval of time into space.) There is even an advantage in not specifying our category precisely: our reasoning will apply to any category in which the axioms are true, and there are, in fact, many such categories (‘topological spaces’, ‘smooth spaces’, etc.).

We begin by stating Brouwer's theorems and by trying to see whether our intuition about continuous maps makes them seem plausible. First we describe the Brouwer fixed point theorems.