Kerodon

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

Proposition 3.3.7.1. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix { X \ar [r]^-{u} \ar [d] & X' \ar [d] \\ S \ar [r]^-{v} & S', } \]

where the vertical maps are Kan fibrations and $v$ is a weak homotopy equivalence. The following conditions are equivalent:

$(1)$

The morphism $u$ is a weak homotopy equivalence.

$(2)$

For every vertex $s \in S$, the induced map of fibers $u_{t}: X_{s} \rightarrow X'_{ v(s)}$ is a homotopy equivalence of Kan complexes.

Proof. Using Corollaries 3.3.6.8 and 3.3.6.5, we can replace $(1)$ and $(2)$ by the following assertions:

$(1')$

The morphism $\operatorname{Ex}^{\infty }(u): \operatorname{Ex}^{\infty }(X) \rightarrow \operatorname{Ex}^{\infty }(X')$ is a weak homotopy equivalence.

$(2')$

For every vertex $s \in S$, the induced map of fibers $u_{s}: \operatorname{Ex}^{\infty }(X)_{s} \rightarrow \operatorname{Ex}^{\infty }(X')_{v(s)}$ is a homotopy equivalence of Kan complexes.

The equivalence of $(1')$ and $(2')$ follows by applying Proposition 3.2.7.1 to the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Ex}^{\infty }(X) \ar [r]^-{\operatorname{Ex}^{\infty }(u) } \ar [d] & \operatorname{Ex}^{\infty }(X') \ar [d] \\ \operatorname{Ex}^{\infty }(S) \ar [r]^-{\operatorname{Ex}^{\infty }(v)} & \operatorname{Ex}^{\infty }(S'); } \]

note that every simplicial set appearing in this diagram is a Kan complex (Proposition 3.3.6.9), the vertical maps are Kan fibrations (Proposition 3.3.6.6) and $\operatorname{Ex}^{\infty }(v)$ is a homotopy equivalence by virtue of Corollary 3.3.6.8. $\square$