Effective groupoid colimits

Let \(C\) be a category with pullbacks and geometric realizations.

1. A geometric realization \(X = \abs{X_{\bullet}} := \colim_{n} X_n\) is called a groupoid colimit if the simplicial object \(X_{\bullet}\colon \simp\catop \to C\) is a groupoid.

2. We say groupoid colimits are effective in \(C\) if for every cartesian transformation \(Y_{\bullet} \to X_{\bullet}\) of groupoid objects, the commutative square

   

   is a pullback square.

3. We say that groupoid colimits are universal if for every morphism \(f\colon A \to B\) in \(C\), the pullback functor \(f^*\colon C_{/B} \to C_{/A}\) preserves groupoid colimits.