Definition
Let \(\Sigma\) be a set. The groupoid of sets \(\mathcal C_\Sigma\) is the groupoid whose objects are elements of \(\Sigma\) and whose morphisms consist only of the identity morphisms. We say that a groupoid \(\mathcal C\) is equivalent to \(\Sigma\) if there is an equivalence of categories \(\mathcal C\simeq \mathcal C_\Sigma\).