nLab
loop groupoid

Definition

For $G$ a group and $BG$ its one-object delooping groupoid, the loop groupoid $ℒBG$ of $G$ is the free loop space object of $BG$.

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

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

Applications

This plays a role in Dijkgraaf-Witten theory.