Kerodon

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Proposition 6.3.6.7. Let $f: X \rightarrow S$ be a universally localizing morphism of simplicial sets. Then $f$ is surjective.

Proof. Let $\sigma : \Delta ^ n \rightarrow S$ be an $n$-simplex of $S$; we wish to show that $\sigma $ can be lifted to an $n$-simplex of $X$. Assume otherwise, so that the inclusion map $\operatorname{\partial \Delta }^{n} \times _{S} X \hookrightarrow \Delta ^ n \times _{S} X$ is an isomorphism. We have a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \times _{S} X \ar [r]^-{\sim } \ar [d] & \Delta ^ n \times _{S} X \ar [d] \\ \operatorname{\partial \Delta }^{n} \ar [r] & \Delta ^ n, } \]

where the vertical maps are weak homotopy equivalences (see Remarks 6.3.6.6 and 6.3.6.4). It follows that the inclusion $\operatorname{\partial \Delta }^{n} \hookrightarrow \Delta ^ n$ is also a weak homotopy equivalence, which is a contradiction (since the relative homology group $\mathrm{H}_{n}( \Delta ^ n, \operatorname{\partial \Delta }^ n; \operatorname{\mathbf{Z}}) \simeq \operatorname{\mathbf{Z}}$ is nonzero). $\square$