I am deeply confused about geometric realizations and finite limits. Suppose that I am working with simplicial sets (I dont need simplicial spaces) that are "good" in the sense of Segal so that I can take fat geometric realization. I have an ordinary pullback square of simplicial sets:
(a) Applying fat geometric realization, do I obtain a homotopy pullback square out of it? If so, will the homotopy pullback square be in the category of topological spaces or compactly generated topological spaces?
(b) I know that applying the ordinary geometric realization will give a pullback square in the category of compactly generated Hausdorff spaces. Does such a pullback square give a long exact sequence of homotopy groups?
In fact, my core question is how we can obtain a long exact sequence of homotopy groups from a pullback square of simplicial sets?