  We show that the Street nerve of a strict $\omega$-category $C$ is a Kan
complex (respectively a quasi-category) if and only if the $n$-cells of $C$ for
$n\geq 1$ (respectively $n> 1$) are weakly invertible. Moreover, we equip
$\mathcal{N}(C)$ with a structure of saturated complicial set where the
$n$-simplices correspond to morphisms from the $n^{th}$ oriental to $C$ sending
the unique non-trivial $n$-cell of the domain to a weakly invertible cell of
$C$.
