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:

$W=X\times_ZY$

(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?

sets; it's only relevant for simplicial spaces (and simplicial objects in other topological bicomplete categories). 2) Fat geometric realisation doesn't require any conditions to exist. 3) Fat realisation preserves pullbacks as ordinary limits, but I can't recall if that's in k-spaces or all spaces. $\endgroup$ – David Roberts Aug 14 '15 at 1:50