[05W2] Lemma 12.4. Let WW be a Segal space. Then the squares W0\displaystyle{{W_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}s0\scriptstyle{s_{0}}W1\displaystyle{{W_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}s0\scriptstyle{s_{0}}d1\scriptstyle{d_{1}}W1\displaystyle{{W_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}s1\scriptstyle{s_{1}}d0\scriptstyle{d_{0}}W0\displaystyle{{W_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}s0\scriptstyle{s_{0}}W1\displaystyle{{W_{1}}}W2\displaystyle{{W_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}d2\scriptstyle{d_{2}}W2\displaystyle{{W_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}d0\scriptstyle{d_{0}}W1\displaystyle{{W_{1}}} are homotopy pullback squares.
[05W3] Proof. Recall that for a Segal space (d0,d2):W2→∼W1×W0W1(d_{0},d_{2})\colon W_{2}\xrightarrow{\sim}W_{1}\times_{W_{0}}W_{1}, so that W2×W1W0→∼(W1×W0W1)×W1W0≈W1W_{2}\times_{W_{1}}W_{0}\xrightarrow{\sim}(W_{1}\times_{W_{0}}W_{1})\times_{W_{1}}W_{0}\approx W_{1} and W0×W1W2→∼W0×W1(W1×W0W1)≈W1W_{0}\times_{W_{1}}W_{2}\xrightarrow{\sim}W_{0}\times_{W_{1}}(W_{1}\times_{W_{0}}W_{1})\approx W_{1}. ∎