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”.

Pullbacks and pushouts are more advanced universal properties, showing some more of the general features that are missing from the universal properties we’ve seen so far. We describe the general idea of a cone over a diagram, and define a limit over a diagram as a universal cone over that diagram, and show that products are in fact limits over a discrete diagram. We define pullbacks as limits over a particular diagram which includes some arrows. We unravel pullbacks in Set, and show that intersections are an example. We also describe the category in which pullbacks are terminal objects, and show how pullbacks can be used to define composable pairs of arrows in the definition of a category. We define pushouts as the dual of pullbacks, and then unravel that as a direct definition. We examine pushouts in Set and show that, given two sets, the square involving their intersection and their union is both a pullback and a pushout. We briefly discuss pushouts of topological spaces as a way of glueing spaces together.

We gather small categories and functors into a category Cat, taking care with size issues to avoid a Russell-like paradox. We consider some functors from Cat to Set, and to the category of graphs and their morphisms. We sketch a free category functor. We look at structures in Cat, much as we have done with other examples of large categories of mathematical structures. We examine terminal and initial objects in Cat, then products and coproducts, and the relationship between (co)products in Cat and those in the categories of posets or monoids. We examine isomorphisms in Cat and show that these exhibit categories with the same arrow structure, such as the cube of factors of 30 and the cube of three types of privilege. We discuss the fact that this concept is overly strict, as it invokes equalities between objects, showing that Cat strains at its dimensions and is trying to expand into higher dimensions. This leads us to the definition of full, faithful, and essentially surjective. We show that full and faithful functors reflect isomorphisms. We define pointwise equivalence and discuss the sense in which this is a version of bijection, not of isomorphism.

An introduction to the concept of the book, the role it aims to play, and the intended audience. Traditionally, mathematics is often thought to progress through levels of difficulty, with category theory being very advanced and therefore only taught to advanced math students. However, in this book we take the view that mathematics is a network of interconnecting ideas, and that, while individual subject areas may progress cumulatively, there is no technical need to study any of them before category theory. So the book is aimed at a wide audience, especially including people who have not studied college-level mathematics.

This chapter begins Part II of the book, in which we build on the basic definition of a category and think about particular types of structure that might be of interest in any given category. This chapter is about how category theory provides a more nuanced approach to sameness, called isomorphism. We define inverses and isomorphisms. We give a sense in which a category treats isomorphic objects as the same. We then study isomorphisms of sets and show that the categorical definition corresponds to the elementary notion of bijection (where “elementary” means “defined with reference to elements”). We then look at isomorphisms of monoids, groups, and partially ordered sets, showing that these are just structure-preserving maps that also happen to be a bijection, and we discuss how these exhibit things with the same structure. We show that the situation is different for topological spaces, as not every bijective continuous map has a continuous inverse. We briefly touch on the idea of isomorphisms of categories, explaining that this is not the best level of sameness of categories. We finish by mentioning further topics: groupoids, categorical uniqueness, and categorification.

The field of quantum research is currently undergoing a revolution. A variety of tools and platforms for controlling individual quantum particles have emerged, which can be utilized to develop entirely new technologies for computation, communication, and sensing. In particular, these technologies will enable applications of quantum information science that can fundamentally change the way we store, process, and transmit information. Exciting theoretical predictions exist for quantum computers, with some proof-of-principle experiments, to perform calculations that would overwhelm the world’s best conventional supercomputers. Quantum research is rapidly developing, and the race is intensifying for quantum technology development, involving some of the high-tech giants. In this chapter we will introduce some key concepts in the materials and devices behind these technological developments. Becoming familiar with these concepts in this first chapter should provide the reader with concrete goals and motivations for studying the quantum methods and tools described in subsequent chapters.

Quantum mechanics is currently the most fundamental theory in use in many disciplines of science and engineering. It is particularly important when one is dealing with nanoscale and atomic-scale systems. However, many phenomena and properties that occur at atomic scales are strange and nonintuitive. There are a number of concepts that simply do not exist in the macroscopic world where we live. Wave–particle duality is one of them. In this chapter, we examine how and when classical particles start behaving as quantum mechanical waves, derive the most important wave equation that quantum particles obey, Schrödinger’s equation, and solve it for the elementary problems of electron waves in given potential energy landscapes. We will also learn how to calculate the expectation values of observables when the wavefunction is known. Schrödinger’s equation will be extensively used throughout the rest of this textbook. More complicated potential energy problems, particularly those relevant to materials and devices, will be dealt with in Chapters 5 and 7, building upon the formulations developed in this chapter.

The band theory of solids provides a general framework with which to understand properties of materials. It not only explains the fundamental differences in electronic structure between insulators, semiconductors, and metals but also provides guidelines for finding optimum materials for specific device applications. For example, a semiconductor with a light effective mass is suited for high-electron-mobility transistors (HEMTs) because the mobility ${\mu }_{\mathrm{e}}$ is inversely proportional to the effective mass, ${\mu }_{\mathrm{e}}=e\tau /m^{*}$, where τ is the scattering time. For developing LEDs and laser diodes, a direct band gap material – i.e., a material in which the conduction-band bottom and the valence-band top occur at the same k – is necessary for momentum conservation since the momentum of photons is negligibly small compared with crystal momenta. In this chapter, after reviewing the basic concepts of atomic and molecular orbitals, bonds and bands, crystal lattices and reciprocal lattices, we provide an overview of the band structure of technologically important materials, including both traditional and emerging materials.