Kerodon

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Warning 4.5.2.26. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X' \ar [r]^-{f'} \ar [d]^{q'} & X \ar [d]^{q} \\ S' \ar [r]^-{f} & S, } \]

where $q$ and $q'$ are Kan fibrations and $f$ is a homotopy equivalence. By virtue of Proposition 3.2.8.1, the following conditions are equivalent:

$(1)$

The morphism $f'$ is a homotopy equivalence of Kan complexes.

$(2)$

For each vertex $s' \in S'$ having image $s = f(s') \in S$, the induced map of fibers $X'_{s'} \rightarrow X_{s}$ is a homotopy equivalence of Kan complexes.

Corollary 4.5.2.25 can be regarded as a generalization of the implication $(1) \Rightarrow (2)$, where we allow $\infty $-categories in place of Kan complexes and isofibrations in place of Kan fibrations. Beware that the implication $(2) \Rightarrow (1)$ does not generalize. For example, we have a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^1 \ar [r] \ar [d] & \Delta ^1 \ar [d]^{ \operatorname{id}} \\ \Delta ^1 \ar [r]^-{\operatorname{id}} & \Delta ^1, } \]

where the vertical maps are isofibrations, the bottom horizontal map is an isomorphism, and the upper horizontal map restricts to an isomorphism on each fiber, but is nevertheless not an equivalence of $\infty $-categories.