Abbreviation: Cat

A \emph{category} is a structure $\mathbf{C}=\langle C,\circ,\text{dom},\text{cod}\rangle$ of type $\langle 2,1,1\rangle$ such that $C$ is a class,

$\langle C,\circ\rangle$ is a (large) partial semigroup

dom amd cod are total unary operations on $C$ such that

$\text{dom}(x)$ is a left unit: $\text{dom}(x)\circ x=x$

$\text{cod}(x)$ is a right unit: $x\circ\text{cod}(x)=x$

if $x\circ y$ exists then $\text{dom}(x\circ y)=\text{dom}(x)$ and $\text{cod}(x\circ y)=\text{cod}(y)$

$x\circ y$ exists iff $\text{cod}(x)=\text{dom}(y)$

Remark: The members of $C$ are called \emph{morphisms}, $\circ$ is the partial operation of \emph{composition}, dom is the \emph{domain} and cod is the \emph{codomain} of a morphism.

The set of objects of $C$ is the set $\mathbf{Obj}C=\{\text{dom}(x)|x\in C\}$. For $a,b\in C$ the set of homomorphism from $a$ to $b$ is $\text{Hom}(a,b)=\{c\in C|\text{dom}(c)=a\text{ and }\text{cod}(c)=b\}$.

Let $\mathbf{C}$ and $\mathbf{D}$ be categories. A morphism from $\mathbf{C}$ to $\mathbf{D}$ is a function $h:C\rightarrow D$ that is a homomorphism: $h(\text{dom}(c))=\text{dom}h(c)$, $h(\text{cod}(c))=\text{cod}h(c)$ and $h(c\circ d)=h(c) \circ h(d)$ whenever $c\circ d$ is defined.

Morphisms between categories are called \emph{functors}.

Example 1: The category of function on sets with composition.

In fact, most of the classes of mathematical structures in this database are categories.

$\text{dom}(\text{dom}(x))=\text{dom}(x)=\text{cod}(\text{dom}(x))$

$\text{cod}(\text{cod}(x))=\text{cod}(x)=\text{dom}(\text{cod}(x))$

$\begin{array}{lr}

f(1)= &1\\
f(2)= &3\\
f(3)= &11\\
f(4)= &55\\
f(5)= &329\\
f(6)= &2858\\
f(7)= &\\
f(8)= &\\
f(9)= &\\
f(10)= &\\

\end{array}$

http://oeis.org/A125696

Schroeder categories

Closed categories

Compact categories

Partially ordered categories

Partial semigroups