Groupoids

Abbreviation: Grpd

Definition

A 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 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 Brandt groupoids.

Examples

Example 1:

Basic results

Properties

Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &2\\ f(3)= &3\\ f(4)= &7\\ f(5)= &9\\ f(6)= &16\\ f(7)= &22\\ f(8)= &42\\ f(9)= &57\\ f(10)= &90\\ \end{array}$

http://oeis.org/A140189

Subclasses

Superclasses

References