Definition 6.29.
For a groupoid object \(\Gg\), we define its classifying object as the geometric realization \(\bB \Gg := \abs{\Gg}\).
If \(\Gg \ltimes X\) is an action of a groupoid object \(\Gg\) on an object \(X \in T\), we define the quotient \(X \quot \Gg\) as the geometric realization of \(\Gg \ltimes X\):
\[X \quot \Gg \quad:=\quad \abs{\Gg \ltimes X} \quad\simeq\quad \colim_{[n] \in \simp\catop} \Gg_n \times_{\Gg_0} X.\]
The cartesian transformation \(\Gg \ltimes X \to \Gg\) induces a map \(X \quot \Gg \to \abs{\Gg} = \bB\Gg\) on geometric realizations, and this defines a functor
\[-\quot\Gg \colon \Act_{\Gg}(T) \to T_{/\bB\Gg}.\]