In this chapter we cover the more advanced universal properties of products and coproducts. We define products directly and then show that this is a terminal object in a more complicated category, the category of cones over the pair of objects in question. This enables us to immediately deduce a uniqueness result. We show that, in Set, cartesian products are categorical products, and examine what uniqueness means. We show that, inside a poset expressed as a category, products are given by least upper bounds. We examine products of posets, and show that this gives us the categories of privilege; we also examine products of monoids and of groups. We then define coproducts as the dual concept, before unraveling this to get a direct definition. We show that, in Set, disjoint unions are coproducts, and we mention decategorification and relationship between categorical (co)products and arithmetic. We show that coproducts in posets are greatest lower bounds, and the coproducts of posets work but coproducts of tosets in general do not, and that coproducts of monoids exist but are much harder than products. We briefly mention coproducts of topological spaces and of categories.

An overview of the abstract side of mathematics, to put us in a good state of mind for category theory. It’s important not to be thinking of math in the common narrow way as numbers, equations, and problem-solving. This chapter will still have little formality. We will discuss the fact that abstraction is important because it takes us to a context in which logic works precisely. Math uses logic and abstraction together, which is how arguments are made very rigorous. We will describe the trade-off, which is that we gain rigor but lose details. However, we will also describe how this in turn gains us wider applicability. Other disadvantages include the fact that it can seem hard and remote from real life, but the difficulty in turn means we are stretching our brains, and the remoteness means we can unify a broader range of examples. We will also describe how math turns abstractions into new concepts and studies them carefully, and how abstraction often increases as we progress through math education, but that doesn’t necessarily mean that abstract math is harder. We will also crucially discuss the fact that abstraction can be fun.

In this chapter we finally give the formal definition of a category. We build up to each part of the definition first, also explaining that definitions of abstract mathematical structures often take the form of data, structure, and properties. We start with the data, which is objects and arrows. Then we give the structure, which is composition and identities. We discuss the fact that arrows can be drawn in many different ways while still giving the same abstract structure. We see that identities are a generalization of reflexivity, and composition is a generalization of transitivity. Finally, we give the properties or axioms, which are the unit laws and associativity of composition. We include a brief mention of size issues, and also describe how associativity can be encapsulated geometrically as a tetrahedron. We discuss the idea of drawing diagrams in category theory, in which we omit identities and composites for efficiency, and introduce the notion of commutative diagrams. The chapter concludes with a discussion of the importance of composition in a category.

Some demystification of the letters used in mathematics, including when we tend to use lower case, upper case, blackboard bold, curly, or Greek letters, and when we use particular letters for particular things, or particular types of thing.

This chapter introduces the idea of studying things via their relationships with other things. We start with family relationships and the idea of depicting relationships with arrows, and building up relationships by composition. We revisit some of the concepts already discussed, and reframe them as types of relationship. This includes symmetry, basic arithmetic, and modular arithmetic. We look at the relationships between different types of quadrilateral and explore how depicting the relationships with arrows is more powerful than using a Venn diagram. We explore diagrams of factors, fixing a number n and then drawing a lattice of its factors and factor relationships among them. We show that abstracting from this gives us a lattice of subsets of the prime factors of n and that a further abstraction gives us lattices of subsets of any sets. We show that this further abstraction enables us to include a much wider variety of examples, including an analysis of interactions between people with different types of privilege. The idea is to start seeing relationships as something quite general, and to start seeing how abstraction gives us structures that are more widely applicable.

We end by taking a step back and summing up some of the main principles of category theory, to make sure that the ideas have not been overwhelmed by the details. We begin with the motivations: the purpose of abstraction, and the idea of gaining structural understanding. We then move on to the general process of actually doing category theory, and end with some aspects of its practice: the idea of morphisms, diagrammatic reasoning, coherence, universal properties, structural awareness, and the nuance to be gained from higher dimensions.

In this chapter we look at the concept of morphisms between categories, using the principle of preserving structure. We give the definition in two ways, one for each of our two approaches to defining categories (by homsets or by graphs). We look at functors between small examples of categories, including functors between posets, monoids, and groups, expressed as categories. We consider functors from small drawable categories and show that they produce a diagram of that shape in the target category. We think about the category consisting of a single non-trivial isomorphism, and see that a functor out of it picks out an isomorphism in the target category. We describe free and forgetful functors, including the free monoid functor. We define the concept of functors preserving and reflecting structure, and show that not all functors preserve epics, but they all preserve split epics. We consider whether the above forgetful functors preserve terminal and initial objects. Further topics include the fundamental group functor, and Van Kampen’s theorem reframed as preservation of pushouts under certain circumstances. We introduce contravariant functors.

A first look at relations more abstractly. We begin by thinking about more relaxed notions of equality and observe that, if we relax the notion too much, then we might get some undesirable consequences. This motivates the concepts of reflexivity, symmetry, and transitivity. We’ll explore these properties for various relations, including informal ones such as “is at school with” and “is a friend of”, and then more formal mathematical ones such as “divides” and “is congruent modulo n”. We introduce equivalence relations, which are those satisfying all three properties, and observe that many relations are not equivalence relations but are nonetheless still interesting and worthy of study. This is to motivate the more relaxed axioms for a category. This chapter starts using more formal notation.

In this chapter we formalize the idea of characterizing things by the role they play in context. First we discuss the difference between characterizing something by role and characterizing it by intrinsic characteristic. We consider “extremities” in some of the small drawable categories we’ve seen before, that is, the places where all arrows start or all arrows end. We make this more precise and more formal in the definition of initial objects. We explore what sort of categories do not have initial objects, including those with an infinite string of composable arrows, non-trivial loops, or disconnected parts. We prove that initial objects are unique up to unique isomorphism. We then define terminal objects and explore them analogously, mentioning the fact that this is the dual concept, although duality is discussed more fully in the next chapter. We then examine terminal and initial objects in some of the categories we’ve seen already: sets, posets, monoids, groups, and different categories of privilege. We observe that we can only describe universal properties in context, as something initial in one category may not be initial in another.

Natural transformations are an appropriate notion of relationship between functors, so this chapter gives us our first glimpse of two-dimensional structures. We first give the definition by feeling our way through abstractly, and then by analogy with homotopies. We define identities and composition for natural transformations, and thus define functor categories. We define the category of presheaves on a category as a particular functor category. We show how to use natural transformations to define cones over a diagram formally. We look at isomorphisms in functor categories, and prove that these correspond to componentwise isomorphisms. We show how to read commutative diagrams “dynamically”. We define equivalence of categories via pseudo-inverses, and briefly mention the relationship with pointwise equivalence. We define horizontal composition of natural transformations, and whiskering, and prove the interchange law, so that we are ready for the concept of a 2-category.

We cover some of the basic building blocks of logic, including logical implication, converse and contrapositive, quantifiers, and their negations.

We explore other large categories based on the category of sets and functions. We begin with monoids, and think of them as sets with extra structure. We introduce the concept of a structure-preserving function, which in this case gives us the definition of a monoid morphism. We check that we can assemble monoids and their morphisms into a category. We do the analogous construction for groups and group homomorphisms. We consider partially ordered sets and show how the structure-preserving functions in this case are order-preserving functions, and we assemble those into a category. We consider some cases of functions that are not order-preserving, illuminating some antagonism that can arise over the theory of privilege. We present the category of topological spaces and continuous maps. We introduce the idea of assembling categories themselves into a category, beginning to think about the idea of structure-preserving maps for categories. These are called functors, and are covered in full in a later chapter. Finally, we touch on matrices, introducing a category where the objects are natural numbers and a morphism from a to b is an a × b matrix. This concludes the interlude of the book.

We can regard all arrows in a category as pointing the other way, and this gives us the dual category. One advantage is that we immediately get a dual version of every construction and every theorem. We begin by exploring some small examples such as categories of factors, and turn all the arrows round to see what the resulting structure looks like. Thus motivated, we make the definition of dual category, and explain that any categorical structure has a dual version which is given by placing that structure in the dual category. We show that in this sense monics and epics are dual, and that isomorphisms are self-dual. We also describe the concept of duals of results, which are found by placing the result in the dual category. We show that the composite of two monics is monic, and that the dual result is that the composite of two epics is epic; we also consider the converse. We show that terminal and initial objects are dual. Finally, we briefly mention how the notion of dual category comes from symmetry in the definition of a category, more easily seen from the definition as an underlying graph with extra structure.

In this final chapter we continue applying the principle of looking at relationships between things, giving more dimensions. We define 2-categories directly, inspired by our understanding of categories, functors, and natural transformations. We revisit the definition of category by homsets, and generalize it to give the definition of 2-category by enrichment. We revisit the definition of category by underlying graph, and generalize it to give the definition of 2-category by underlying 2-graph. We define the two types of duality for 2-categories, and discuss the appropriate notions of sameness for different dimensions of morphism in a 2-category. We define monoidal categories as 2-categories with only a 0-cell, and show the dimension shift that is analogous to the one for monoids and categories. We discuss the issue of strictness and weakness, give the unit triangle and associativity pentagon, and discuss coherence. We discuss degeneracy and the Eckmann–Hilton argument, leading to braidings. We give an introduction to how research proceeds up the dimensions, giving an overview of various approaches. The chapter becomes less formal and rigorous as we end with a taste of open research.

Introduction to the idea that math is all relative to context, as things behave differently in different contexts. This chapter has a little more formality. We begin by discussing the myth that math is rigid and fixed, and introduce the idea that, on the contrary, it is contextual, and has great flexibility coming from the ability to move between different contexts. As an example, we look at the taxi-cab metric and examine what circles are in this context, and what ? (“pi”) is, where ? is defined as the ratio between the circumference and diameter of any circle. We then look at some different contexts in which 1 + 1 can be considered to be something other than 2, including the “n-hour clocks”, that is, arithmetic modulo n, and the zero world in which everything is zero. This is to open our thinking to the idea that different things can be true in different contexts.

A first example of a large category of mathematical structures. This means that, instead of looking at an individual structure as a category, we look at all structures of a certain type, and appropriate morphisms between them, and express that as a category. Sets and functions are an essential starting point of mathematics, and one of the fundamental motivating examples of category theory. We start by giving an account of functions that is more aligned with higher level mathematics, and is possibly different from how functions are usually treated in high school. We also examine the total number of possible functions between a given set of inputs and a set of outputs. We then define the identity function, and composition of functions, and check the unit and associativity laws, to show that sets and functions do indeed form a category, which we call Set. Finally, we introduce the idea of sets with extra structure, and the important difference between expressing properties of functions at the level of elements, or at the level of objects and morphisms in the category Set. The latter is the idea of expressing things “categorically”.