Starting Category Theory

The following sections are included:

• Dedication
• Preface
• Acknowledgments
• About the Author
• Contents

In this chapter, will look at the basic building blocks of category theory: categories, functors, and natural transformations.

As we mentioned in the introduction, whenever we introduce a new concept, we begin by giving a rigorous definition followed immediately by an intuitive explanation. So, in case the abstract definition is hard to understand at first, read on and come back to it after seeing the examples.

In this chapter, we study the notions of representable functors and of universal property, and the Yoneda lemma. These ideas turn category theory from a mere language into a mathematical theory of its own.

Some of these concepts, such as universal properties, will be very abstract — if you are a beginner in category theory, the content of this chapter is likely to be the most abstract mathematics you have ever seen. It may be that initially, one has no clear mental picture for this abstract ideas. If this is the case, start with the more concrete examples later on in the chapter: they usually help a lot. You can also first check out the more concrete Sections 3.1 and 3.2 in the following chapter (skipping the references to Yoneda) and then come back to this material for the general theory.

In Section 1.4, we remarked how natural transformations are a generalization of equivariant maps and how invariant maps are a special case of equivariant maps when one of the actions is trivial. Natural transformations, when one of the functors is trivial, form diagrams that look like “cones,” and an “optimal” (universal) choice of cone is called a limit. This construction gives one of the most fruitful ideas in category theory, used in almost all areas of mathematics.

In this chapter, we look at another cornerstone concept in category theory: adjoint functors. Whenever a functor is not invertible, very often we have a “canonical choice of coming back,” but different from a section or a retraction. This notion is central to the theory of monads (Chapter 5) and to the theory of closed monoidal categories (Section 6.5).

An alternative way of reading this book, which could be helpful to some readers, is to read the following chapter on monads before this one and to come back here having in mind that monads give rise to adjunctions.

This chapter is dedicated to monads and comonads, which sit at the intersection of mathematics and computer science. Here are the main definitions, which we interpret in several ways in the following sections…

In this last chapter, we study monoidal categories. They are one of the most fruitful ideas of category theory, with applications as diverse as quantum information theory, algebraic geometry, and probability.

Monoidal categories would require an entire book of their own, or even several, just for an introduction. The material here, however, should at least be enough to get you started.

While using monoidal categories in mathematical practice very often saves time and effort (especially using string diagrams, see Section 6.1.2), developing the necessary theory often involves lengthy proofs and rather large diagrams. For reasons of space, we cannot include all these proofs in their entirety here — we instead provide the basic ideas and references for further learning.

The following sections are included:

• Conclusion
• Bibliography
• Index