Kerodon

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Proposition 3.4.1.11 (Transitivity). Suppose we are given a commutative diagram of simplicial sets

3.44
$$\begin{gathered}\label{equation:composite-homotopy-pullback} \xymatrix@R =50pt@C=50pt{ Z \ar [r] \ar [d]^{h} & Y \ar [r] \ar [d]^{g} & X \ar [d]^{f} \\ U \ar [r] & T \ar [r] & S } \end{gathered}$$

where the right half of (3.44) is a homotopy pullback square. Then the left half of (3.44) is a homotopy pullback square if and only if the outer rectangle is a homotopy pullback square.

Proof. By virtue of Proposition 3.1.7.1, the morphism $f$ factors as a composition $X \xrightarrow {w_ X} X' \xrightarrow {f'} S$, where $f'$ is a Kan fibration and $w_{X}$ is a weak homotopy equivalence. Set $Y' = T \times _{S} X'$, so that $g$ factors as a composition $Y \xrightarrow { w_{Y} } Y' \xrightarrow {g'} T$ where $g'$ is a Kan fibration. Since the right half of (3.44) is a homotopy pullback square, the morphism $w_{Y}$ is a weak homotopy equivalence. Applying Proposition 3.4.1.2, we see that both conditions are equivalent to the requirement that the induced map $Z \rightarrow U \times _{T} Y' \simeq U \times _{S} X'$ is a weak homotopy equivalence. $\square$