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 $\pi\_0$).



Connectivity after Geometric Realization?
Yes it is a surjection on $\pi\_0$, because each component of $|Y|$ has at least one component of $Y\_0$.


Beyond that there are no restrictions. For instance, you can get any homotopy type for $|X|$ and $|Y|$ and any homotopy type for the map between them with $X\_0$ and $Y\_0$ just one point, as long as you ask that $\pi\_0(|X|)$ and $\pi\_0(|Y|)$ are trivial.


(I'm taking "simplicial space" to mean a simplicial object in the category of topological spaces, say the compact Hausdorff ones.)


