Does the suspension functor preserve fibrations?

Question

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?

Answer 1

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.