The language in this is overwhelming

[nickretallack]

I started reading this hoping that it would be a good way for me to understand category theory when most explanations are full of overwhelming amounts of math lingo that's difficult for me to picture. I was doing fine until this part in 3.3:

Let’s characterize these ordered sets as categories. A preorder is a category where there is at most one morphism going from any object 𝑎 to any object 𝑏. Another name for such a category is “thin.” A preorder is a thin category. A set of morphisms from object 𝑎 to object 𝑏 in a category 𝐂 is called a hom-set and is written as 𝐂(𝑎, 𝑏) (or, sometimes, Hom𝐂(𝑎, 𝑏)). So every hom-set in a preorder is either empty or a singleton. That includes the hom-set 𝐂(𝑎, 𝑎), the set of morphisms from 𝑎 to 𝑎, which must be a singleton, containing only the identity, in any preorder. You may, however, have cycles in a preorder. Cycles are forbidden in a partial order. It’s very important to be able to recognize preorders, partial orders, and total orders because of sorting. Sorting algorithms, such as quicksort, bubble sort, merge sort, etc., can only work correctly on total orders. Partial orders can be sorted using topological sort.

My eyes glazed over reading this. What do all these words mean? I continued reading, hoping the rest would be simpler, but then I got to 3.5. Argh.

In other words, we have the hom-set 𝐌(𝑚, 𝑚) of the single object 𝑚 in the category 𝐌. We can easily define a binary operator in this set: The monoidal product of two set-elements is the element corresponding to the composition of the corresponding morphisms. If you give me two elements of 𝐌(𝑚, 𝑚) corresponding to 𝑓 and 𝑔, their product will correspond to the composition 𝑓 ∘ 𝑔. The composition always exists, because the source and the target for these morphisms are the same object. And it’s associative by the rules of category. The identity morphism is the neutral element of this product. So we can always recover a set monoid from a category monoid. For all intents and purposes they are one and the same.

I mean, I kind of get it after reading this paragraph 5 times, but this phrasing is really dense and difficult for me to follow. Perhaps if it was spread out a bit and had examples?

I also don't understand the exercises in this chapter, and I'm not sure how to know if I did them right because there doesn't appear to be an answer key.

[BartoszMilewski]

There is a tradeoff between how slowly one can explain new things and the resulting size of the book. This book is already thick, I'm afraid. Reading every paragraph 5 times is not a bad idea. This is how you read math books. When I started reading MacLane, it took me weeks to understand the first few paragraphs. Eventually you catch up, though, so don't give up.

[frgomes]

@nickretallack : Another strategy is "just keep reading", without caring about too much with every possible interpretation behind every single line. Just keep reading. "You just need to get used to it." -- said my first Physics teacher at uni, when explaining concepts far ahead of lessons of Calculus which should preceed lessons of Physics... but they didn't. "Just get used to it."... just keep reading... and the coin will eventually drop later. Then, if you read the book once again you will be able to understand what it is about in every detail, since you will have knowledge captured from "future" lessons.

Unfortunately, certain subjects are difficult to explain in a sequential manner, since the knowledge mankind accumulated on those subjects was not built linearly, but by trial and error, by injecting lessons learned from the future and into weak theoretical basis built in past, in a process of continuous feedback loop, refining again and again the previous iterations.