### Table of Contents

## Groupoids

Abbreviation: **Grpd**

### Definition

A \emph{groupoid} is a category $\mathbf{C}=\langle C,\circ,\text{dom},\text{cod}\rangle$ such that

every morphism is an isomorphism: $\forall x\exists y\ x\circ y=\text{dom}(x)\text{ and }y\circ x=\text{cod}(x)$

##### Morphisms

Let $\mathbf{C}$ and $\mathbf{D}$ be Schroeder categories. A morphism from $\mathbf{C}$ to $\mathbf{D}$ is a function $h:C\rightarrow D$ that is a \emph{functor}: $h(x\circ y)=h(x)\circ h(y)$, $h(\text{dom}(x))=\text{dom}(h(x))$ and $h(\text{cod}(x))=\text{cod}(h(x))$.

Remark: These categories are also called \emph{Brandt groupoids}.

### Examples

Example 1:

### 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)= &2\\ f(3)= &3\\ f(4)= &7\\ f(5)= &9\\ f(6)= &16\\ f(7)= &22\\ f(8)= &42\\ f(9)= &57\\ f(10)= &90\\

\end{array}$