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## Categories

% Note: replace “Template” with Name_of_class in previous line

Abbreviation: **Cat**

### Definition

A ** 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

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

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

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

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

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 ** morphisms**, $\circ$ is the partial operation of

**, dom is the**

*composition***and cod is the**

*domain***of a morphism.**

*codomain*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 $\text{Hom}(a,b)=\{c\in C|\text{dom}(c)=a\text{ and }\text{cod}(c)=b\}$.

##### Morphisms

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: $h(\text{dom}(c))=\text{dom}h(c)$ and $h(c\circ d)=h(c) \circ h(d)$ whenever $c\circ d$ is defined.

### Examples

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

### Basic results

### Properties

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$ $\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$

### Subclasses

[[...]] subvariety

[[...]] expansion

### Superclasses

[[...]] supervariety

[[...]] subreduct

### References

^{1)}F. Lastname,

**, Journal,**

*Title***1**, 23–45 MRreview

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