Example 2

Let $G = (S,F, o, t)$ be a groupoid. There exists a unique operation of G on the set S, relative to Id$_S$. It is given by $\varphi (f)(a) =b$ if $f\in G_{a,b}$. The orbits for this operation are the orbits of G in the sense defined on p. 162.

Let G be a groupoid, let S be the set of its vertices, let X be a set, and let $p: X\rightarrow S$ be a mapping. Let $\varphi$ be an operation of G on X relative to $p$. One says that the groupoid G operates without monodromy on X if, for every point $a\in S$ and every element $f\in G_a,\varphi (f)$ is the identity of $X_a$. If the groupoid G is transitive, it is enough that this be so for one point of S.

Let $a$ be a point of S and suppose that one has $\varphi (f) =$ Id$_{X_a}$ for every $f\in G_a$. Let $b$ be a point of S belonging to the orbit of $a$ and let $g$ be an arrow of G joining $a$ to $b$. For every $f\in G_b$, Int($g$)$(f) =gf g^{-1}$ is an element of $G_a$; hence one has Id$_{X_a}=\varphi (gf g^{-1}) =\varphi (g)\varphi (f)\varphi (g)^{-1}$, so that $\varphi (f) =$ Id$_{X_b}$. Let $b$ be a point of S belonging to the orbit of $a$ and let $g,g'$ be arrows of G joining $a$ to $b$; then $g'g^{-1}$ is a loop at $a$, whence $\varphi (g'g^{-1}) =$ Id$_{X_a}$ and $\varphi (g) =\varphi (g')$.