Igor wrote:

> John, could you please recommend some paper/book/blog as a good introduction to such systematical methods?

Any introduction to category theory contains these insights; two of my favorites are these free ones:

* Tom Leinster, _[Basic Category Theory](https://arxiv.org/abs/1612.09375)_, Cambridge Studies in Advanced Mathematics, Vol. 143, Cambridge University Press, 2014. (An introduction.)

* Emily Riehl, _[Category Theory in Context](http://www.math.jhu.edu/~eriehl/context.pdf)_, Dover, New York, 2016. (More advanced.)

_However,_ it's hard to get these insights without carefully studying several of these books and _also_ talking to an expert several times a week for a long time. That's why I'm teaching this 6-month-long course!

> Progressing so far through the book/course already brought a lot of value, but conditions like this one for monoidal preorders make me feel uneasy. Handwavingly I knew that this is just another monotonicity condition, but why exactly this one, and how the form to express it was chosen? - questions like this bothered me this week.

The solution is to ask me those questions! I'm not getting enough questions in this course.

I guess you're asking about this condition:

\[ x \le x' \textrm{ and } y \le y' \textrm{ imply } x \otimes y \le x' \otimes y' .\]

I wanted everyone to think about this condition, so I posed this puzzle:

**Puzzle 66.** What's another way to state this last condition using ideas we've already discussed? Hint: it's a property of the function \\(\otimes : X \times X \to X\\).

The property is that \\(\otimes : X \times X \to X\\) is _monotone_.

Remember, given two preorders \\(X\\) and \\(Y\\) there's an important way to make \\(X \times Y\\) into a preorder, where we say \\((x,y) \le (x',y') \\) iff \\(x \le x'\\) and \\(y \le y'\\). This is the **product in the category of preorders** - when you learn a bit more category theory you'll know what that means and why it's important. (Or maybe you already do!)

So, when \\(X\\) is a preorder, so is \\(X \times X\\), and it makes a lot of sense, when studying _monoids that are also preorders_, to demand that \\(\otimes : X \times X \to X\\) be monotone. This makes \\(X\\) into a **monoid in the category of preorders**.

It's very similar to how an "associative algebra" is a monoid in the category of vector spaces, or a topological group is a group in the category of topological spaces. If these examples are unfamiliar, I apologize: they don't really matter, they just indicate that in math we're often faced with the task of hybridizing two structures, say A's and B's — and a good approach is often to define an "A in the category of B's".

This can be done systematically; it's called **internalization**, and we're seeing an example when we're discussing monoidal preorders. There is a lot more to say about it:

* nLab, [Internalization](https://ncatlab.org/nlab/show/internalization).

When you know internalization, you can just say:

**Definition.** A monoidal preorder is a monoid in the category of preorders.

and this _implies_ the condition

\[ x \le x' \textrm{ and } y \le y' \textrm{ imply } x \otimes y \le x' \otimes y' .\]

This is very nice.

However, that nLab page may be difficult to understand without a lot of experience in category theory! That's why I'm teaching this course.

Ask questions....