The mathematical and physical construction of quantum mechanics is undeniably beautiful, but as hinted in the Introduction, why the universe should ultimately be quantum mechanical is mysterious. Of course, empirical science can never answer the question of “Why?” definitively, but only establish the rules that govern Nature through experiment. Nevertheless, there were several points in our discussion of the motivation for the Hilbert space, the Born rule, or the Dirac–von Neumann axioms that seemed to be completely inexplicable and potentially inconsistent with the guiding principles we used. In this chapter, we survey a few of these points from an introductory, modern perspective. Quantum mechanics works, makes precise predictions, and agrees with experiment, but what quantum mechanics is is still very much an open question.

We’ve come a long way from the fundamental mathematical properties of linear operators to the profound physical interpretations of them. We have finally developed the necessary background for deriving the fundamental equation of quantum mechanics, the Schrödinger equation.

In the previous chapter, we studied the consequences of rotations and angular momentum in three spatial dimensions, building up to the topic of this chapter: the quantum mechanics of the hydrogen atom. Hydrogen is, of course, the lightest element of the periodic table and consists of a proton and an electron bound through electromagnetism. Our goal for studying this problem is to determine the bound-state energy eigenstates, just like we did with the infinite square well and harmonic oscillator. These energy eigenstates will then tell us how the proton and electron are positioned with respect to one another in space, as well as the energy levels and how energy is transferred when the hydrogen atom transforms from one energy level to another. As always, our goal is to diagonalize the Hamiltonian $\hat{H}$; that is, determine the states $|\psi ⟩$ and energies E such that

This chapter is the beginning of the interlude, in which we give a tour of various parts of mathematics that are often taken as prerequisites for studying category theory. Here, instead, we will introduce them as examples of categories, in a series of shorter chapters. We begin in this chapter by revisiting some of the structures from the first part of the book and showing how they are all in fact categories. This includes symmetries, equivalence relations, diagrams of factors, and the natural numbers.

Another example of doing things categorically, that is, moving away from elements and expressing structure in terms of morphisms in a category. We examine the asymmetry in the definition of a function, leading to the definitions of injective and surjective functions. We examine examples and non-examples from life, which include people stepping on your foot, and people experiencing homelessness; mathematical examples and non-examples include functions from the integers to the integers that add 1 or multiply by 2, and also the empty function. We define monics and epics as categorical versions of injective and surjective functions, and show that in Set they actually correspond to injections and surjections. We show that isomorphisms are necessarily monic and epic, but that a monic and epic morphism is not necessarily an isomorphism in other categories. We examine this in the category of monoids. We conclude by mentioning some further topics: density, the use of monics to define subobjects, and the issues of generalizing monics and epics to higher dimensions.

This chapter brings together all that we’ve done in one of the pinnacles of abstraction in category theory. First, we revisit the sense in which a category sees isomorphic objects as the same, and show that our argument from Chapter 14 is in fact an isomorphism in Set between some particular sets of morphisms. We then show how this arises from some particular types of functor called representable functors. We then go up another level and introduce the Yoneda embedding as a functor from our base category to the category of presheaves on it, and we show that it is full and faithful. We describe the principle behind the Yoneda Lemma, and then state the Yoneda Lemma. Although we have all the technology required for the proof, we stop just short of giving it. We end the chapter with a brief discussion of Mac Lane’s comment that all concepts are Kan extensions.

An overview of what the point of category theory is, without formality, and an overview of the contents of the book. We will present category theory as “the mathematics of mathematics”, so first we explain what aspects of mathematics we are focusing on. We present mathematics as starting from abstraction, as a way of elucidating analogies between situations, finding connections between them, and unifying them. Category theory is then a rigorous framework for making analogies and finding connections between different parts of mathematics. It focuses on relationships between things, rather than on intrinsic characteristics, and uses those relationships to put objects in context rather than treat them in isolation. Once the framework has been set up, we have, among other things, a way to express more nuanced notions of “sameness”, and a way to characterize things by the role they play in that context. Category theory also works at many different levels, so we can zoom in and out and study details close up, or broad contexts with more of an overview.

A slightly more formal treatment of topological spaces, including open sets, the definition of continuous functions via open sets, and homotopy.

A discussion of the role of pattern spotting in mathematics. This chapter still has little formality. The idea is that, in math and also in life, we might make abstract versions of things, then spot patterns in them or in common between different situations, and then ask if those patterns are caused by some abstract structure. We begin with some pattern spotting in number squares, and introduce the arithmetic of the 12-hour clock and look at the patterns on its addition table. We discuss patterns as analogies between situations, and discuss the relationship between visual patterns and abstract patterns, one example being the visual patterns made by multiples of 3 on a number square (or multiples of other fixed numbers). We introduce systemic power structures in society as another example of patterns caused by abstract similarities. We discuss how abstraction can help us spot patterns, with one example beingthe analysis of contentious arguments in society and politics, many of which follow a similar pattern.

A brief treatment of Russell’s Paradox and how this leads to the idea of a hierarchy of different sizes of totalities.

An alphabetized reference list of technical mathematical words that come up to help the reader who might need reminding. Also a list of notation.

Ordered sets can be thought of as categories with particularly tidy arrow structures. We will first look at totally ordered sets. We will give the definition of an ordered set as a special kind of category, and then unravel that to give the direct definition as a set with extra structure, the latter being the definition that is usually given when this topic is covered before category theory. Examples include the natural numbers ordered by size. We will explore features that cause a category to fail to be an ordered set, looking at the features visually and also formally. We then generalize to partially ordered sets, again giving the definition as a special kind of category and then as a set with structure. Examples include the factors of any fixed number, ordered by the relation “is a factor of”.

In this chapter we zoom in and look at small examples of mathematical structures that we can express as categories. We start with small drawable examples, that is, finite categories, and also sets expressed as discrete categories. We then move on to monoids expressed as categories with only one object, and unravel that to give the elementary definition of a monoid as a set with structure. We describe the monoid of natural numbers as generated by a single arrow consisting of a non-trivial loop. We explain the “dimension shift” involved in forgetting the single object and focusing on the set of morphisms as the underlying data. We then move on to groups expressed as categories with only one object in which every morphism has an inverse, and then unravel that as an elementary definition. Examples include modular arithmetic and groups of symmetries. We include some broader context about group theory. Finally, we introduce the idea of points and paths in a space, touching on some of the concepts of topology and homotopy, without technicality (we include some technical details in an Appendix).

This chapter eases us from informal ideas into formal mathematics. We motivate that move, develop more formal approaches to structures we’ve already seen, acclimatize ourselves to the formalism, and see what we get out of it. We argue that formal mathematical language and notation are to do with both efficiency and abstraction, as concise notation can help us to package up multiple concepts into a single unit that we can then handle as a new object. One example is repeated addition turning into multiplication, and then repeated multiplication turning into exponentiation. We revisit metrics and express them more formally. We also cover some basic formal logic, which is how formal mathematics is built up securely, including logical implication, converses, and logically equivalent ways of stating them. We also revisit modular arithmetic and show how to express it more formally.