Remark 3.3.8.3. If $f: X \rightarrow S$ is a Kan fibration of simplicial sets, then every vertex $s \in S$ determines a Kan complex $X_{s} = \{ s\} \times _{S} X$. One can think of the construction $s \mapsto X_{s}$ as supplying a map from $S$ to the “space” of all Kan complexes. Roughly speaking, one can think of Theorem 3.3.8.1 as asserting that this “space” itself behaves like a Kan complex. We will return to this idea in §5.6.
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