Lemma 6.33.
Let \(X \in T\) be an object such that \(X \to *\) is an effective epimorphism. Let \(\Gg \ltimes X\) be an action of a groupoid object on \(X\). Then the action is regular if and only if the quotient \(X\quot\Gg\) is the terminal object of \(T\).
Proof
First assume that \(X\quot\Gg \simeq *\). Then the map \(X \twoheadrightarrow X\quot\Gg\) is equivalent to the map \(X \to *\). It follows that the shear map \(\shear\colon (\Gg \ltimes X) \to \check{C}(X \to *)\) is an isomorphism, being defined by applying Čech nerves to these two equivalent morphisms.Conversely, assume that \(\Gg \ltimes X\) is regular and that \(X \twoheadrightarrow *\) is an effective epimorphism. The map \(X\quot\Gg \to *\) is obtained from the higher shear map \(\shear\colon (\Gg \ltimes X) \to \check{C}(X \to *)\) by passing to colimits. It thus suffices to show that each higher shear map \(\shear_n\colon \Gg_n \times_{\Gg_0} X \to X^{n+1}\) is an isomorphism. Both its source and target are groupoid objects with object of objects \(X\). By the Segal condition, their terms in degree \(n\) are the \(n\)-fold iterated fiber products of their terms in degree \(1\) over \(X\), and \(\shear_n\) is the corresponding iterated pullback of \(\shear_1\). Since \(\shear_1\) is an isomorphism by regularity, every \(\shear_n\) is an isomorphism.