Definition 6.132.
A pretopos \(C\) is called bounded if it is small and every \(X \in C\) is truncated.
A pretopos \(C\) is hypercomplete if the following conditions are satisfied:
It admits colimits of Kan simplicial objects, meaning simplicial objects whose matching maps satisfy the internal Kan lifting condition.
If \(X_{\bullet} \to X\) is a trivial Kan fibration, meaning a Kan fibration whose relative homotopy objects are all trivial, then \(\abs{X_{\bullet}} \simeq X\).