Groupoid

Remarks

A groupoid is small if its objects form a set. For each object $x$, the morphisms $x \to x$ form a group, the vertex group at $x$.

Example: let $\sim$ be an equivalence relation on a set $X$. Take $X$ as the set of objects, with exactly one morphism $x \to y$ when $x \sim y$ and none otherwise. Composition exists by transitivity, identities by reflexivity, and the inverse of $x \to y$ is the morphism $y \to x$, which exists by symmetry.

Non-example: the category with objects $0, 1$ and one non-identity arrow $0 \to 1$, the poset $0 \le 1$. There is no morphism $1 \to 0$, so the arrow $0 \to 1$ has no inverse.

A group $G$ gives a groupoid with one object whose morphisms are the elements of $G$, and every one-object groupoid arises in this way.