Groupoid action

Let \(\Gg\) be a groupoid object in \(T\), and let \(X \in T\) be an object. An action of \(\Gg\) on \(X\) consists of the following data:

1. A groupoid object \(\Gg \ltimes X\) in \(T\);

2. An isomorphism \((\Gg \ltimes X)_0 \simeq X\);

3. A cartesian natural transformation \(c\colon \Gg \ltimes X \to \Gg\) of simplicial objects in \(T\).

We let \(\Act_{\Gg}(T) \subseteq \Grpd(T)_{/\Gg}\) denote the full subcategory spanned by the \(\Gg\)-actions in \(T\).