nLab loop groupoid

Definition

For $G$ a group and $\mathbf{B}G$ its one-object delooping groupoid, the loop groupoid $\mathcal{L}\mathbf{B}G$ of $G$ is the free loop space object of $\mathbf{B}G$.

This is the groupoid whose objects are the elements of $G$, and whose morphisms are of the form

$$g \stackrel{h}{\to} h^{-1} g h \,.$$

Applications

This plays a role in Dijkgraaf-Witten theory.