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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?

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    $\begingroup$ 1) You don't need to think about 'goodness' for simplicial 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 ♦
    Commented Aug 14, 2015 at 1:50
  • $\begingroup$ Welcome to MO Alex? $\endgroup$
    – Paul Fabel
    Commented Aug 14, 2015 at 3:47
  • $\begingroup$ @DavidRoberts Do you have a reference for the statement that the fat realization preserves pullbacks? $\endgroup$
    – archipelago
    Commented Aug 14, 2015 at 15:38
  • $\begingroup$ @archipelago I've still got to either find it or prove it myself... $\endgroup$
    – David Roberts ♦
    Commented Aug 16, 2015 at 1:22
  • $\begingroup$ @archipelago apologies if you were the same person who asked me this before (I can't find the relevant discussion here on MO), but if you haven't seen it, then the statement is given without proof as 2.23 in arxiv.org/abs/math/0701916, namely that sTop -> Top/||pt|| preserves finite limits. $\endgroup$
    – David Roberts ♦
    Commented Aug 17, 2015 at 2:24

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No, there's no reason for this to be true without additional hypotheses, e.g. fibrancy hypotheses on the maps $X \to Z$ and $Y \to Z$. Suppose $Z$ is connected and has at least two $0$-simplices in the same connected component and that $X$ and $Y$ are two distinct such $0$-simplices. Then $W$ is empty, but the homotopy pullback is the based loop space $\Omega Z$.

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    $\begingroup$ And: pullback squares that aren't homotopy pullback squares don't give you long exact sequences. $\endgroup$
    – Qiaochu Yuan
    Commented Aug 15, 2015 at 15:36

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