Definition 4

A groupoid is a category each arrow of which is invertible.

Let G be a groupoid and let $a$ be a vertex of G. The monoid $G_{a,a}$ is a group, denoted by $G_a$ and called the isotropy group of G at $a$.

Let $a,b$ be vertices of G and let $f\in G_{a,b}$ be an arrow joining $a$ to $b$. The mapping $g\mapsto f gf^{-1}$ is a group isomorphism of $G_b$ onto $G_a$, denoted by Int($f$). If $f$ and $f'$ are composable arrows of G, one has Int($f f'$) $=$ Int($f$)$\circ$ Int($f'$) ; the inverse of the isomorphism Int($f$) is the isomorphism Int($f^{-1}$).

A morphism of groupoids $\varphi : G\rightarrow G'$ is a morphism of categories. Moreover, $\varphi$ is an isomorphism of groupoids if and only if it is an isomorphism of categories.

Let $\varphi : G\rightarrow G'$ be a morphism of groupoids. For every vertex $a$ of G, the mapping $f\mapsto \varphi (f)$ of $G_a$ into $(G')_{\varphi(a)}$ is a group homomorphism denoted by $\varphi_a$. In particular, one has $\varphi (e_a) =e_{\varphi(a)}$.

For every arrow $f$ of G$,\varphi (f^{-1})$ is the inverse of $\varphi (f)$. If $f$ is an arrow of G joining a vertex $a$ to a vertex $b$, one thus has, for every element $g\in G_b$, the relation Int($\varphi (f)$)$(\varphi (g)) =\varphi$(Int($f$)$(g)$).