Lemma 6.4.18 (Pushouts as geometric realizations). Let \(D\) be an \(\infty \)-category with finite coproducts and geometric realizations. Given a span \(X \xleftarrow {f} A \xrightarrow {g} Y\), there exists a simplicial object \(B_{\bullet }(X,A,Y)\) with \[ B_n(X,A,Y) := X \sqcup A^{\sqcup n} \sqcup Y. \] Its geometric realization is the pushout: \[ \abs {B_{\bullet }(X,A,Y)} \cong X \sqcup _A Y. \]
Proof. TO DO. This is an instance of the Bousfield–Kan formula for colimits. □
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