## Category Theory

Category theory is the mathematics of mathematics. Okay, so then what is Mathematics? It’s the logical study of how logical things work. Thus category theory is the logical study of how logical things work.

Category makes logical connections between things

A family tree is an example of category theory. It provides a way of representing relationships.

Category theory and composition explained using family trees. #ylj18 pic.twitter.com/f4Wn4doIV9

— Geoffrey Huntley (@GeoffreyHuntley) May 21, 2018

# Object and morphisms

Communicative squares

#ylj18 Commutative squares with Super Smash Bros pic.twitter.com/LILzxGJM6Q

— Andrae Muys (@etymon) May 21, 2018

Factors of 42

Mind slightly detonated. Visualizing number factors relationships in higher dimensions, and deriving an abstraction leading to general principle. “Maths is getting fed up with washing the dishes and inventing a dishwasher” - @DrEugeniaCheng #ylj18 pic.twitter.com/cjLfNwEaqv

— Christopher Biggs (@unixbigot) May 21, 2018

Cube of privilege

A cube of privilege #YLJ18 pic.twitter.com/ZAcX6KuhKv

— Vaibhav Sagar (@vbhvsgr) May 21, 2018

# The principle of Sameness

```
8 + 1 = 1 + 8
2 x 5 = 5 x 2
(8+5) +5 = 8 + (5+5)
```

@DrEugeniaCheng says that doing math with numbers is boring, and we should look at using math in the kitchen where one equation doesn't necessarily equal the other (except cake, cake always works) #ylj18 🍰 pic.twitter.com/3prdodCgaN

— Amy 👩💻🐺 (@Amys_Kapers) May 21, 2018

The concept of invertibility - “if I go over here can I go back?”

# Universal Properties

Things can be isomorphic in one category but not another.

If you freeze water, it goes back to water. If you freeze an egg and

Whole categories can be isomorphic if the arrows look the same. The first row is isomorphic because the arrows go the same way, the bottom row isn’t isomorphic.

🌶️💯🌶️ #ylj18 pic.twitter.com/e7IMDjUw5x

— Vaibhav Sagar (@vbhvsgr) May 21, 2018

# Visual Representation

Friend map

With this one slide you can introduce folks to category theory. Start at the bottom and do why why why until you get to the top. #ylj18 pic.twitter.com/n9xeq9iQkb

— Geoffrey Huntley (@GeoffreyHuntley) May 21, 2018

Cycles

Category theory: helping to explain why losing weight is complicated #ylj18 pic.twitter.com/65MgUS45AK

— Andrew Herron (@_spyder) May 21, 2018

We often ignore the side-effects of what we do, like the "waste" shown in red here. That's one reason we're in trouble. But category theory can make this process of ignoring into a functor! Then, left and right adjoints can help us "un-ignore": https://t.co/wUhc3YjKds pic.twitter.com/4Zp0B9gsoz

— John Carlos Baez (@johncarlosbaez) May 21, 2018

# References

- https://arxiv.org/abs/1803.05316