# Category Theory (Apr.2021. updating continuously)

## Materials

## Definition of a category

- Objects and arrows are there.
- There should be identity arrows.
- Arrows should be composited.
- Associativity holds.

Terminology: domain and codomain

$\mathrm{dom}(f)\xrightarrow{f} \mathrm{cod}(f)$

## Functor

- A Functor is a mapping between categories.
- It’s like an arrow from category to category.
- It preseves compositions $F(g\cdot f) = F(g)\cdot F(f)$.
- Identitiy arrows are mapped to identity arrows.

## Isomorphism

In a category $C$, two objects $A$ and $B$ are called **isomorphic** If therea is arrows $f$ and $g$ such that $f:A\rightarrow B,\ g:B\rightarrow A$ and $f\cdot g =1_A,\ g\cdot f=1_B$.
Such $f$ and $g$ are called isomorphism.

## Product category

Imagine simple tuple which can be defined from two categories.

- Projection functors p1(C,D) = C p1(f,g) =f

## Dual category

dual category = opposite category = arrows reversed. denoted as A over asterisc (A^*)

## Free category

## Pass

- arrow category
- slice category

## 1.4

$$ {\mathrm Sets}_{\mathrm{fin}} $$

is the category of all finite set sand functions between them. <- very large!

#### Poset and Preordered set

Poset = partially ordered set = reflexivity, transitivity, and antisymmetry. Preorder = reflexivity and transitivity.

#### Rel

$$ Rel $$ objects = sets and arrows binary relations aRb is true.

$$ Pos $$ is large also.

#### Functor

#### Cat

$$ Cat $$ the category of al vategorieCatnd functors (lbstrefrenzitaet)

#### Monoid

### 1.5 isomorphism (category)

$$ A \cong B $$

f:A -> B is isomorphism if g:B->A exist s.t. gf = 1A fg=1B

this is one of mophism.

Final object

Mrphism ist arrow

Multi Terminal Objekt are All isomorphic

In Terms of se Theorie, Singleton ist Onkyo Terminal Objekt

preorder set is similar with category without parallel arrows

category product is the most good candidate. most good means unique arrow

we can ask is there a product in this category?

## initial morphism

initial property dual -> Universal morphism

## memo

- Universal properties: basical
- Natural transformation: map from a functor to another functor
- Corn: continuously drive a structure
- Limit: No more natural transformation in the corn.Generalized category sproduct?