CatGloss

Consider xXx \in X as a representative of its orbit OxO_x. Because the translation groupoid is equivalent to its skeleton, we must have \(\Hom_{\cat{sk}\cT_GX}(O_x,O_x)\cong\Hom_{\cT_GX}(x,x) =: G_x,\) the set of automorphisms of xx. This group consists of precisely those gGg \in G so that gx=xg \cdot x = x. In other words, the group $\Hom_{\cT_GX}(x,x)$ is the stabilizer GxG_x of xx with respect to the GG-action. Note that this argument implies that any pair of elements in the same orbit must have isomorphic stabilizers. As is always the case for a skeletal groupoid, there are no morphisms between distinct objects. In summary, the skeleton of the translation groupoid, as a category, is the disjoint union of the stabilizer groups, indexed by the orbits of the action of GG on XX.