### Part II Category Theory – 9. What Is Category Theory?

- Category theory is the mathematics of mathematics. Whatever mathematics does for the world, category theory does for mathematics. It’s the study of the structures that hold mathematics up. Math is vast enough to need its own system of organization, which is why category theory was born.

### 10. Context

- The character of a number depends on the context in which it is placed. For example, 5 can be a counting number, a prime, or part of a fraction. Category theory highlights the context. This chapter contains many examples.

### 11. Relationships

- A recipe is used to show the importance of relationships. If a recipe calls for a cup on one ingredient and two cups of another, it doesn’t matter what size cup you use. In math, a function takes one thing and produces something else. The function itself defines the relationship between the two things. Category theory represents relationships with diagrams. A category in mathematics starts with a set of objects and a set of relationships between them.

### 12. Structure

- The baking metaphor used here is the recipe for baked Alaska. It contains a cake on the bottom and meringue on the top, both of which insulate the ice cream in the middle and prevent it from melting while baking. Without this structure the ingredients arranged any other way would come out a sloppy mess. Category theory examines structures as well to determine which are necessary. Once just the outer walls of a building go up, just about any building can pass for a parking garage. The chapter then uses a cake with an internal pattern as another example.

### 13. Sameness

- Here we start with a cookie recipe that produces cookies that are vegan, sugar-free, low-fat, and raw. The point is that the more substitutions you make, the farther you get away from the original concept. Toothbrushes and toilet brushes are both brushes, but that is where the sameness ends. It all depends on context. We then meet notions of sameness in math like similar triangles, equilateral triangles, the topological sameness of a coffee cup and a donut. In category theory, things only count as more or less the same if you can reverse the process. Going from point A to point B cannot be reversed, for example, if you have a one-way street. You can freeze and unfreeze water and get the same thing, but you can’t do so with an egg. The sets {1,2,3} and {cat, dot, banana} don’t seem the same until you realize they both are composed of three things.

### 14. Universal Properties

- We start with a recipe for fruit crisp, which has the universal properties of whatever fruit you come up with topped with a mixture of flour, sugar, and butter. In category theory, an example would be that you can add zero to any number and get the same number. It’s like the fact that the glass slipper
*only*fit Cinderella. A property that characterizes an object as unique is a universal property. An example would be that Eugenia is the only female category theorist in South Yorkshire. Category theory seeks to characterize things by the role they play and to think up roles and look for something that plays those roles.

### 15. What Category Theory Is

- Recall that math is all built on logic and nothing else. This is very different from other fields like science, which is evidence-based. Math doesn’t use evidence. Intuition and imagination have no place. The Trinity of Truth is laid out here. It includes your
*Knowledge*of the outside world, your*Beliefs*from your inside, and your*Understanding*, which holds the first two together. - Mathematics in schools can seem like an autocratic state with strict, unbending rules that seem arbitrary. Teachers need to believe that knowledge is power, but understanding is more powerful power. Don’t just show students the rules and keep them in the dark about reasons. Math is certainly an area that needs much more
*illumination*. Make sure that teachers you know, especially math teachers, are doing what they can to bring more light to what they teach. Thanks, Eugenia for bringing more light to my understanding of mathematics.

### Eugenia Cheng

- Eugenia has previously been on the mathematics faculty at the University of Chicago and is the Scientist in Residence at the School of the Art Institute of Chicago where she lives.

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