Remark 4.8.5.21. In the situation of Proposition 4.8.5.20, suppose that $f$ is a Kan fibration. Then assertions $(a)$ and $(b)$ can be reformulated as follows:
- $(a')$
The Kan fibration $f$ is $0$-full if and only if, for each vertex $y \in Y$, the fiber $X_{y} = \{ y\} \times _{Y} X$ is nonempty.
- $(b')$
For $n \geq 1$, the Kan fibration $f$ is $n$-full if and only if, for every vertex $x \in X$ having image $y = f(x)$, the set $\pi _{n-1}( X_{y}, x)$ consists of a single element.
See Corollary 3.2.6.8 (and Variant 3.2.6.9 for the case $n = 1$).