Groupoid

A groupoid is a nonempty set $G$ with a unary operation ${}^{-1}: G \to G$, and a partial function $* : G \times G \to G$. Let $a, b, c \in G$, the following properties must hold:

• Associativity in the following sense:
  • If $a * b$ and $b * c$ are defined, then $(a * b) * c$ and $a * (b * c)$ are defined and $(a * b) * c = a * (b * c).$
  • $(a * b) * c$ is defined if and only if $a * (b * c).$
  • If $(a * b) * c$ is defined, then $(a * b) * c = a * (b * c)$ and $a * b$ and $b * c$ are defined.
• Identity in the following sense:
  • If $a * b$ is defined, then $a^{-1} * a * b = b$ and $a * b * b^{-1} = a.$
• Inverses in the following sense:
  • $a * a^{-1}$ and $b * b^{-1}$ are always defined.

Remarks

In other words, a groupoid is a category where every morphism is invertible.