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Let $X_{\bullet}$ be a simplicial set and let $\Sigma X_{\bullet}$ denote its simplicial suspension. If $X_{\bullet} \to Y_{\bullet}$ is a fibration, then is $\Sigma X_{\bullet} \to \Sigma Y_{\bullet}$ also a fibration?

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The answer is no.

It is enough to provide a counterexample in spaces, in which case we may take the double cover of the circle $S^1 \rightarrow S^1$. The suspension would have fibers consisting of one point (at the points we coned off in the suspension) or two points generically. But fibers in a fibration are homotopy equivalent, so this fails.

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