Observation about groupoid-action on a unique object.

Let $G_0$ be the set of half-lines of the plane with common origin $O$, it has a topology induced by the usual biiection with the circle $S^1$. Consider the structure of indiscrete order on $G_0$, this is a groupoid $G$ , a morphism of $G$ is a pair $(r, s)$ of half-lines and is defined the angle from $r$ to $s$ (we fix the counterclockwise direction as positive), then $G$ acts on $G_0$ by geometric rotation.

In other way, if you have a (topological) group $G_0$, define the small indiscrete groupoid $G$ with object class $G_0$, and $G_0(x, y)= \{yx^{-1}\}$ (this is the comma $*\downarrow G_0$, where $\star$ is the unique object of $G_0$), $G$ act on $G_0$ in obvious way (composition).

It seem that this make a functor form the groups to the indiscrete-groupoid actions on a pointed set (the special point is the unity), we have a obvious reverse functor too.

Observation about groupoid-action on a unique object.

Let $G_0$ be the set of half-lines of the plane with common origin $O$, it has a topology induced by the usual biiection with the circle $S^1$. Consider the structure of indiscrete order on $G_0$, this is a groupoid $G$ , a morphism of $G$ is a pair $(r, s)$ of half-lines and is defined the angle from $r$ to $s$ (we fix the counterclockwise direction as positive), then $G$ acts on $G_0$ by geometric rotation.

In other way, if you have a (topological) group $G_0$, define the small indiscrete groupoid $G$ with object class $G_0$, and $G_0(x, y)= \{yx^{-1}\}$ (this is the comma $*\downarrow G_0$, where $\star$ is the unique object of $G_0$), $G$ act on $G_0$ in obvious way (composition).

It seem that this make a functor form the groups to the groupoid (transitive faithful)-actions on a pointed set (the special point is the unity), we have a obvious reverse functor too.