Sets, maps and the map composition will be talked about in this note.

## Maps and Diagrams

An **object** in finite category means a finite set or collection.

A **map** $f$ in a category consists of three things:

- set $A$ called
**domain**of the map - set $B$ called
**codomain**of the map - A rule assigning to each element $a$ in domain, an element $b$ in codomain.

A map where domain and codomain are the same object is called

endomap.If $\forall a \in A$, $f(a) = a$, this means $f$ is an

identity map, or simply $1_A$.

External Diagram is a scheme to keep track of domain and codomain, without indicating all the detail in map. Each external diagram can correspond to some map:

$f$ that has $A$ be

**domain**and $B$ as**codomain**:

$$

\require{AMScd}

\begin{CD}

A @>{f}>> B

\end{CD}

$$$g$ being an endomap on $A$:

$$

\require{AMScd}

\begin{CD}

A @>{g}>> A

\end{CD}\\

A^{\huge{\circlearrowright}^{\Large g}}

$$$1_A$ being an identity map:

$$

\require{AMScd}

\begin{CD}

A @>{1_A}>> A

\end{CD}\\

A^{\huge{\circlearrowright}^{\normalsize 1_A}}

$$

The composition of two maps, with forms of

$$

\require{AMScd}

\begin{CD}

X @>{g}>> Y @>{f}>> Z

\end{CD}

$$

can be written into $f \circ g$, which is in internal diagram form like

$$

\require{AMScd}

\begin{CD}

X @>{f \circ g}>> Z

\end{CD}

$$

A singleton set is a set with exactly one element, and we can donate that set with $\boldsymbol{1}$.

A point of set $X$ is a map $\boldsymbol{1} \longrightarrow X$.

## Composing Maps and Laws

Identity law was defined as $f: A\longrightarrow B, 1_B \circ f\equiv f \circ 1_A \equiv f$.

Associative law was defined as $f : A\longrightarrow B, g:B \longrightarrow C, h: C \longrightarrow D, h\circ(g\circ f) \equiv (h\circ g)\circ f$.