Corollary 3.5.1.24 (051E)

Corollary 3.5.1.24. Let $n$ be an integer and suppose we are given a homotopy pullback square of simplicial sets

3.67

\begin{equation} \begin{gathered}\label{equation:homotopy-pullback-square-connectivity} \xymatrix@R =50pt@C=50pt{ X \ar [r] \ar [d]^{f} & X' \ar [d]^{f'} \\ Y \ar [r]^-{g} & Y'. } \end{gathered} \end{equation}

If the morphism $f'$ is $n$-connective (in the sense of Variant 3.5.1.6), then $f$ is also $n$-connective. Moreover, the converse holds if $g$ is surjective on connected components.

Proof. Using Proposition 3.1.7.1, we can reduce to the case where $f$ and $f'$ are Kan fibrations. In this case, our assumption that (3.67) is a homotopy pullback square guarantees that for every vertex $y \in Y$, the induced map of fibers $X_{y} \rightarrow X'_{ g(y) }$ is a homotopy equivalence of Kan complexes (Example 3.4.1.4). The desired result now follows from criterion of Proposition 3.5.1.22 (together with Remark 3.5.1.23). $\square$