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Let $X_{\bullet}$ be a simplicial space and denote the (non-fat!) geometric realization by $\lvert X_{\bullet} \rvert$.

Does this geometric realization of simplicial spaces preserve finite limits?

This is well-known to be true for simplicial sets instead of simplicial spaces. Moreover, the fat geometric realization preserves finite limits up to homotopy and the fat-free geometric realization preserves pullbacks and products on the nose (see Proposition 8 and 9 here for references to proofs). So there might be some hope.

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    $\begingroup$ Just to summarize: We are looking at the map $|\lim X|\rightarrow \lim |X|$. The map should be a continuous bijection (Forgetting the topology should reduce this to the case of simplicial sets). So if there is a counterexample, it has to be a ugly one, either this map is not proper or the spaces are not CGHaus. $\endgroup$
    – HenrikRüping
    Commented Aug 8, 2013 at 15:15
  • $\begingroup$ Does spaces mean the category of topological spaces, or the category of compactly generated Hausdorff spaces? If the latter, then you are done by Proposition 8 + the preservation of the terminal object. $\endgroup$
    – David Carchedi
    Commented Aug 8, 2013 at 21:24
  • $\begingroup$ I might miss something here, but using the singular functor $Sing$ on each top. space $X_n$, you obtain a bisimplicial set $X_{\bullet\bullet}$. Then by Eilenberg-Zilber: $\left|DX_{\bullet}\right|\cong\left|X_\bullet\right|$, where $DX_n:=X_{nn}$ denotes the diagonal. Since $Sing$ is a right adjoint, it preserves limits, and since $DX$ is a simplicial set, it's geometric realization preserves finites limits. $\endgroup$
    – Roman Bruckner
    Commented Aug 9, 2013 at 7:48
  • $\begingroup$ @RomanBruckner: But going to the singular simplicial set changes everything up to homotopy equivalence. So, don't you only get that finite limits are preserved up to homotopy this way? $\endgroup$
    – Ulrich Pennig
    Commented Aug 9, 2013 at 7:52
  • $\begingroup$ Umh, yes. That's what I've missed... $\endgroup$
    – Roman Bruckner
    Commented Aug 9, 2013 at 8:03

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To avoid leaving this question open:

Assuming we work in the category of compactly generated spaces, geometric realization commutes with pullbacks.(It's crucial that we use the compactly generated product.) The proof is basically the same as for simplicial sets. A reference in the space-case is Corollary 11.6 of Peter May's book 'The Geometry of Iterated Loop Space'. The terminal object is preserved as well, so the claim follows by abstract nonsense, since a functor preserves all finite limits iff it preserves pullbacks and the terminal object.

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