Definition

The general theory of categories. A category is a tuple $$\mathcal{C} = (O,A,\circ,dom,cod)$$ where

$ O $ is a collection of objects.
$ A $ is a collection of arrows. These are also called morphisms.
$ dom $ and $ cod $ assigns to each arrow $ f $ an object. For $ dom $ the object is called the domain of the arrow. For $ cod $ it is called the co-domain. For $ dom\ f = A $ and $ cod\ f = B $ we write $ f : A \rightarrow B $. The collection of arrow with $ A $ as domain and $ B $ as codomain is written as $ \textbf{C}(A,B) $.
$ \circ $ is a binary operator that assigns to each pair of arrows $ f : A \rightarrow B $ and $ g: B \rightarrow C $ a composite arrow $ g \circ f : A \rightarrow C $. Note that in order for this operator to be well defined, we must have that $ cod\ f = dom\ g $. For arrows we have the following two axioms:
1. Associativity: For any objects $ A,B,C,D \in \mathcal{C} $ and any arrows $ f : A \rightarrow B,g : B \rightarrow C, h : C \rightarrow D \in \mathcal{C} $ we have that $$ f \circ (g \circ h) = (f \circ g) \circ h $$
2. Identity: We have one arrow $ id \in \mathcal{C} $ for which for every $ f \in \mathcal{C} $ we have that $$ f \circ id = id \circ f = f $$

Associativity: For any objects $ A,B,C,D \in \mathcal{C} $ and any arrows $ f : A \rightarrow B,g : B \rightarrow C, h : C \rightarrow D \in \mathcal{C} $ we have that $$ f \circ (g \circ h) = (f \circ g) \circ h $$

The axioms closely resembles those for a a monoid.

Example - The category Set

The category Set has sets as objects and total functions between these sets as arrows. That is

$ O $ is a collection of sets. Or the set of all sets.
$ A $ are total functions between sets. This is how we normally define a function between sets, eg. $$ f : \mathbb{R} \rightarrow \mathbb{R} $$ for the function that maps elements real numbers to real number.
The composition of two arrows are as expected when composing two functions. That is for $ f : A \rightarrow B $ and $ g : B \rightarrow C $ we get $$ g \circ f : A \rightarrow C $$ where for $ a \in A $ we have $ g(f(a)) \in C $.
For any set $ A $ we have $ id_A $ - that is a total function that maps elements from $ A $ to itself.

Example - The categories of algebras

For any algebra we have a category where the objects are instances of this algebra, and where the arrows are homomorphisms between these instances. For example for monoids we have the category Mon for which:

The objects are monoids.
The arrows are monoid homomorphism. That is a function $ \varphi : M_1 \rightarrow M_2 $ from a monoid to a monoid for which $$ \varphi(a \star b) = \varphi(a) \diamond \varphi(b) $$ where $ a,b \in M_1 $ and $ \varphi(a),\varphi(b) \in M_2 $.

Example - Partial Ordering

A partial order $ \leq $ on a set $ P $ is a relation that is reflexive, transitive and anti-symmetrical. That is

reflexive: For $ a \in P $ we have that $ a \leq a $
transitive: For $ a,b,c \in P $ we have that $ a \leq b \land b \leq c \Rightarrow a \leq c $.
anti-symmetrical: For $ a,b \in P $ we have that $ a \leq b \land b \leq a \Rightarrow a = b $.

This gives rise to a category with partial ordered sets as objects and order preserving (monotone) total functions as arrows.

Example - The category 0

The category $ 0 $ has no objects and no arrows.

Example - The category 1

The category $ 1 $ has one object and one arrow - the arrow $ id $ from the one objects to itself.

Example - The category 2

The category $ 2 $ has two objects, two identity arrows and one extra arrow from the one object to the other.

Example - The category 3

The category $ 3 $ has three objects, three identity arrows and three arrows between objects. Say the objects are $ A,B,C $. Then the three arrows are $$f : A \rightarrow B, h : A \rightarrow C, g : B \rightarrow C$$

Example - A monoid

As stated earlier the axioms for arrows within a category resembles those of a monoid. For a monoid $ (M,\cdot,e) $ we can form a category where

We have one object, namely the set $ M $.
The arrows are elements of $ M $.
$ e $ is represented by the identity arrow.
The operator $ \cdot $ is represented by the composition of arrows, that is $ \circ $.

This goes the other way too: Every category with a single object gives rise to a monoid. For example the category $ 1 $ could be interpreted as the monoid $$ (\{1\},\cdot,1) $$ where the operator is normal multiplication.

Example - A Partial Ordered Set

Given a partial ordered set $ (P,\leq) $ we can construct a category where the objects are elements of $ P $ and there is a arrow from objects $ p $ and $ p' $ if and only if $ p \leq p' $. Now we have that

Given two arrows $ f : p \rightarrow p',g : p' \rightarrow p'' $ we must have $ g \circ f : p \rightarrow p'' $. Hence if $ p \leq p' $ and $ p' \leq p'' $, we have that $ p \leq p'' $.
We have $ id : p \rightarrow p $ in this category. Hence we are ensured to have $ p \leq p $.

We disregard anti symmetry. This is called a preorder. Every preorder set gives rise to a category.