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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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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. –  HenrikRüping Aug 8 '13 at 15:15
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. –  David Carchedi Aug 8 '13 at 21:24
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. –  Roman Bruckner Aug 9 '13 at 7:48
@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? –  Ulrich Pennig Aug 9 '13 at 7:52
Umh, yes. That's what I've missed... –  Roman Bruckner Aug 9 '13 at 8:03

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