If you want to get a general feeling of what category theory is all about and whether or not it is something you want to learn more about, this is the right book for you! The analogies between everyday actions, like making food, and category theory are easy to understand.

### PART I: Mathematics

I am going to argue that maths is defined by the techniques it uses to study things, and that the things it studies are determined by those techniques. (p. 5)

In order to build a machine to do something rather than doing it yourself, you have to understand that thing at a different level. (p. 28)

I still think it’s a good idea to know the principles of something that you’re using all the time, so that you’re less at its mercy when it goes wrong (p. 45)

In mathematics, ‘harm’ means ‘causing a logical contradiction’. If something doesn’t cause a logical contradiction, you might as well do it. (p. 113)

As long as your new idea doesn’t cause a contradiction, you are free to invent it. (p. 115)

As I have mentioned before, in maths the basic ingredients are called axioms and the process of breaking something down into its basic ingredients is called axiomatisation. (p. 117)

Maths is about removing the human judgement from things, so that everything proceeds just by logic. (p. 125)

If you start with nothing, you get nothing. So maths isn’t about ‘absolute truth’ after all (p. 129)

So, in fact, rationality is a sociological notion. (p. 157)

### PART II: Category theory

it emphasises their relationships with other objects, as the main way of placing them in context. (p. 184)

The relationships are actually called ‘morphisms’ to allow for the fact that they might not be quite like relationships. (p. 186)

You can wonder whether there’s one ‘special object’ in your world that somehow encapsulates tons of important information all by itself. That is, a sort of barometer object, a litmus test object, a benchmark object, an Erdős-like figure. Mathematicians call this a universal property. (p. 188)

The first rule is reflexivity, which says that everyone is related to themselves. The second rule is symmetry, which says that if A is related to B then B is related to A. The third and last rule is transitivity, which we already saw in Chapter 4. This rule says that if A is related to B and B is related to C, then A is related to C. (p. 194)

A category in mathematics starts with a set of objects, and a set of relationships between them. (p. 201)

Understanding something from only one point of view is far too restrictive. (p. 213)

The only useful equations are those that tell us two different ways of doing something are ‘somehow the same’. (p. 228)

I think a lot of our personal scientific knowledge is just that – knowledge that we believe because somebody we trust has told it to us. We have taken it on trust, or on authority. (p. 274)

‘Because we’ve proved it’ is not a satisfactory answer, from a human point of view. (p. 276)

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