Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 4.3.7.3. Let $f: K \rightarrow X$ and $q: X \rightarrow S$ be morphisms of simplicial sets. Then:

  • If $q$ is a left fibration, then the induced map

    \[ X_{/f} \rightarrow X \times _{ S} S_{/ (q \circ f)} \]

    is a Kan fibration.

  • If $q$ is a right fibration, then the induced map

    \[ X_{f/} \rightarrow X \times _{ S} S_{(q \circ f)/} \]

    is a Kan fibration.

Proof. Apply Proposition 4.3.7.2 in the special case $K_0 = \emptyset $. $\square$