Suppose that I have a map of simplicial spaces,

$ f: X_* \to Y_*$,

and that I know that the map on zero spaces $f_0: X_0 \to Y_0$ is n-connected. Can I conclude anything about the connectivity of the map of geometric realizations?

$ |f|: |X| \to |Y|$

Are there any reasonable conditions I can place on the simplicial spaces X and Y that would allow me to conclude something along these lines? I'm especially interested in knowing when the map is 0-connected (i.e. a surjection on $\pi_0$).