Category Theory and E: Category Theory Basics (English)

Category Theory Basics

Category

A category $\mathcal{C}$ consists of two parts:

• a class of this category's objects: $\text{Obj}_\mathcal{C}$; and
• for every two objects in this category $\mathcal{C}$, such as $A$ and $B$, there is a set of morphisms: $\text{Hom}_\mathcal{C}(A,B)$.

Among them, morphisms satisfy the following requirements:

• for each object, there exists at least one morphism called identity. $\forall A\in \text{Obj}_\mathcal{C}\text{, }\exists I_A \in \text{Hom}_\mathcal{C}(A,A)$
• we can compose morphisms. $\forall f \in \text{Hom}_\mathcal{C}(A,B) \land g \in \text{Hom}_\mathcal{C}(B,C)\text{, }\exists h=gf\in \text{Hom}_\mathcal{C}(A,C)$
• the composition of morphisms satisfies the associative law. $f(gh)=(fg)h$
• the combination of morphisms and the identity is the morphism itself. $f I_A = f = I_A f$

For the convenience of understanding, I give an example:

I have a magic book called "Transformation". With this book, I can transform one animal into another. All the animals that I can change are "objects", and all the magic that changes their shape is "morphisms".

Obviously, my magic has several characteristics:

• I always have a magic called "do nothing", which can turn an animal into... itself. Turn a mouse into a mouse, turn a cat into a cat.
• If I have a magic that can turn a cat into a mouse, and another magic that can turn a mouse into a dog, then I definitely have a magic that can turn a cat into a dog: I just need to combine the two magics, first turn the cat into a mouse, then turn it into a dog.
• When I write spells, I just need to write "cat cat into mouse mouse, mouse mouse into dog dog, dog dog into duck duck" in order. I can write the second half of the spell first and then the first half, or write the first half first and then the second half, but as long as the final writing is in this order, there is no difference.
• If I do nothing and then use magic, or if I use magic and then do nothing, there is no difference from using the magic directly.

General Properties (Universal Properties)

If there exists an object X in a category $\mathcal{C}$ such that there is a morphisms from X to every object in the category, then we call it the initial object. On the other hand, if there exists an object such that there is a morphisms from every object in the category to it, then we call it the terminal object. If an object is both the initial and terminal object, we call it the zero object.

A universal property in a category is the initial or terminal object in that category.

$(\forall A \in \text{Obj}_\mathcal{C} \text{, } \exists ! I_{XA} \in \text{Hom}_\mathcal{C}(X,A)) \lor (\forall A \in \text{Obj}_\mathcal{C} \text{, } \exists ! I_{AX} \in\text{Hom}_\mathcal{C}(A,X))$

For ease of understanding, I give an example:

Among the magic recorded in my magic book of transformation, there are several special magics. These magics can turn all the animals mentioned in the book into sheep. Of course, these are separate magics (they are not a magic), such as dog-sheep spells and cat-sheep spells. But the common feature of these sheep spells is that they can turn any animal mentioned in this book into sheep! (But I don't guarantee that I can turn the sheep back!) So the sheep is a terminal object and a universal property.

Self-Constructed Structures

Abbreviation

Given a category $\mathcal O$, if we can define a subcategory $\mathcal O_A$ (i.e. any object and morphism in the subcategory $\mathcal O_A$ are respectively objects and morphisms in the category $\mathcal O$), and there exists a universal property $A$ in this subcategory. We call this subcategory $\mathcal A$.

Tower Structure

Consider the category $\mathcal T$ where the objects are all morphisms from morphisms to morphisms. Then the morphisms in this category are naturally morphisms from morphisms to morphisms, so the morphisms in this category are also objects in this category.

We can consider the following method to construct this category:

0. Given a category $\mathcal O$, we call this category the 0th meta order category.
1. Construct a category $\mathcal T_1$ where the Objects ($\text{Obj}_\mathcal{T_1}$) are the morphisms in the category $\mathcal O$. We call this category the 1st meta order category.
2. Construct a category $\mathcal T_2$ where the Objects ($\text{Obj}_\mathcal{T_2}$) are the morphisms and objects in the category $\mathcal T_1$. We call this category the 2nd meta order category.
3. Repeat step 2.
4. We can stop at a certain step or proceed to $\mathcal T_\infty = \mathcal T$.

Note that the above step 0 can be omitted:

1. Given a category $\mathcal T_1$, we call this category the 1st meta order category.
2. Construct a category $\mathcal T_2$ where the Objects ($\text{Obj}_\mathcal{T_2}$) are the morphisms and objects in the category $\mathcal T_1$. We call this category the 2nd meta order category.
3. Repeat the previous step.
4. We can stop at a certain step or proceed to $\mathcal T_\infty = \mathcal T$.

We will call these two constructions methods tower construction. We say that an object $A \in \text{Obj}_\mathcal{T_\infty}$ is of absolute a-th meta order (absolute a-th meta order) when it belongs to the a-th meta order category, but does not belong to the (a-1)-th meta order category ($|\text{MetaOrder}(A)| = a$ if $A \in \text{Obj}_\mathcal{T_a} \land A \not \in \text{Obj}_\mathcal{T_{a-1}}$).

We say that an absolute a-th meta order object $A \in \text{Obj}_\mathcal{T_\infty}$ is of (b-a+1)-th meta order relative to an absolute b-th meta order object $B \in \text{Obj}_\mathcal{T_\infty}$ ($b>a$) ($\text{MetaOrder}_B(A) = |\text{MetaOrder}(B)| - |\text{MetaOrder}(A)| +1$).

Clearly, given an absolute 1-th meta order object $R$, any absolute n-th meta order object $A \in \text{Obj}_\mathcal{T_\infty}$ is of n-th meta order relative to $R$. We call $R$ the cornerstone, and when a cornerstone object is given, when we say that an object $A$ is of n-th meta order, we are saying that $A$ is of n-th meta order relative to the cornerstone object.

To facilitate understanding, I give two examples:

Example: Magic Book 3

I am a professor at a magic academy. My rich knowledge of magic... the objects of my transformation spells are no longer just animals. I can even turn one magic into another, realizing the mutual transformation of magic... ah, I still remember the afternoon when the academy asked me to organize it into a magic book after mastering this skill for the first time...

0. I randomly picked up a magic book from the bookshelf, the cover of which read "Complete Animal Transformation Spells - Ultimate Sheep Special Edition" ($\mathcal O$). Oh! This is the first magic textbook I wrote, this is a good start...
1. I spread out the sheepskin paper and began writing my new magic book. Let me see the first two magic spells in this book... Oh! It's Cat Cat turning into a dog spell and Duck Duck turning into a dog spell. Hmph... How simple! I waved my feather pen and wrote - Cat Cat and Duck Duck turn into a dog! This is a brand new magic that was not in the previous book (because the magic in the previous book was not effective on both animals)! Such a simple task is really funny! I made it in accordance with the same law, and quickly finished this magic book - "Composite Animal Transformation Guide - Grade One" ($\mathcal T_1$). With this book, I no longer need to refer to the previous "Complete Animal Transformation Spells - Ultimate Sheep Special Edition".
2. I pull out a new piece of sheepskin paper, this time, I want to write down some magic that is slightly more difficult for those high-grade magic students-but not too difficult. I looked at the just-written "Composite Animal Transformation Guide-Grade 1", and a lazy idea came to mind. I looked at the previously written " Cat Cat and Duck Duck all turned into Dog Dog spell" and "Pig Pig and Rat Rat all turned into Dog Dog spell", wrote ... Yes! It's "Cat Cat Duck Duck Pig Pig Rat Rat all turned into Dog Dog spell"! I made it like this ... I quickly finished this magic book- "Composite Animal Transformation Guide-Grade 2", and put it next to "Composite Animal Transformation Guide-Grade 1". ($\mathcal T_2$ is two books together. The absolute meta order of the magic that appears in the second volume is 2. The absolute meta order of the magic that appears in the first volume is 1.)
3. I continued to write a book in this way. Of course! I will always refer to all the books I have written since grade 1! This can greatly improve the efficiency of the work. For example, in the first book, I wrote "Cat Cat and Duck Duck all turned into Dog Dog spell", in the second book, I wrote "Lion Lion Tiger Tiger Frog Frog Otter Otter all turned into Dog Dog spell", so naturally, I have to write in the third book "Cat Cat Duck Duck Lion Lion Tiger Tiger Frog Frog Otter Otter all turn into Dog Dog spell"!
4. Such an easy task! In that afternoon, I continued to write almost countless magic books according to the same routine!

Note that the professor's method of constructing new magic will result in the last magic book, necessarily containing the true ultimate sheep-changing spell that turns all animals into sheep. But it will not include the true ultimate dog-changing spell that turns all animals into dogs, because dogs are not a terminal objects in $\mathcal O$.

Notice that, fundamentally, the "cat cat and duck duck both become dog dog spell" is actually a magic that converts the "cat cat becomes dog spell" and the "duck duck becomes dog spell" into each other. So we can naturally create a new magic: "cat cat becomes dog spell" (or "duck duck becomes dog spell") and "cat cat becomes dog spell" (or "duck duck becomes dog spell") "added" together to obtain a "cat cat and duck duck both become dog dog spell". This "cat cat and duck duck both become dog dog spell" obviously has the same structure as the "cat cat becomes dog spell" "duck duck becomes dog spell" because for each "cat cat becomes dog spell" and "duck duck becomes dog spell" I can find a unique "cat cat and duck duck both become dog dog spell".