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

  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 $$ f \circ (g \circ h) = (f \circ g) \circ h $$

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

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:

Example - Partial Ordering

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

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

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

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