For the purposes of this question a topological space will mean a compactly generated weak Hausdorff space, though I am actually somewhat flexible on what category of topological spaces we use. I would be interested to know if things are any different if we consider the category of $\Delta$-generated spaces, for example.
A simplicial topological space $X$ is a functor from the combinatorial simplex category to the category of topological spaces. Such a thing has a geometric realization, which is new topological space $|X|$. Categorically it is defined as the coend, which is given equationally as:
$$ |X| = (\bigsqcup X_n \times \Delta^n )/ \sim $$
This is a standard construction in topology and much of it is reviewed here.
A map of simplicial spaces $X \to Y$ (i.e. a natural transformation) is a levelwise weak equivalence if each map $X_n \to Y_n$ is a weak homotopy equivalence.
I am curious to know:
If a map of simplicial spaces is a levelwise weak equivalence, is the induced map on geometric realizations a weak homotopy equivalence?
A similar result appears to hold true if we use semisimplicial spaces instead of simplicial spaces (these are defined exactly like simplicial spaces but with the category $\Delta$ replaced by $\Delta_+$, the category of finite non-empty ordered sets and strictly order preserving maps). I am interested in the simplicial case.
There are many well-known partial results along these lines, where further assumptions are placed on the simplicial spaces. I will describe them briefly. I like to think of these sorts of results as arising from the existence of a Quillen adjunction between topological spaces with some model structure and simplicial topological spaces with the Reedy version of that model structure.
If the geometric realization fits into such a Quillen adjunction (it is a left adjoint), then it automatically preserves weak equivalences (in that model structure) between cofibrant objects.
The most well-known example of this is when we use the h-model structure (similar to the Str\om model structure). Here the "weak equivalences" are the ordinary homotopy equivalences and the cofibrations are the closed Hurewicz cofibrations. In this case the geometric realization does appear to give a Quillen adjunction between the Reedy model structure on simplical spaces and the h-model structure on topological spaces. What this means is that if the simplcial spaces are h-Reedy cofibrant (which means the inclusions $X(\partial \Delta^n) \to X_n$ are closed Hurewicz cofibrations) then geometric realization will send a levelwise homotopy equivalence to a homotopy equivalence. This is essentially the condition called "proper" and it is implied by what Segal called "good".
I believe that a similar statement also holds using the "mixed" model structure on topological spaces, in which the fibrations are the Hurewicz fibrations, but where the weak equivalences are the weak homotopy equivalences. This compliments the above result and says that if the simplicial spaces where Reedy cofibrant in the mixed model structure, then levelwise weak homotopy equivalence are sent to weak homotopy equivalences.
Still, I am interested in the general case. Partly I am just curious about the general case and if there is a counter-example, but also partly because I am considering a particular example where the simplicial spaces involved are not obviously (to me) Reedy cofibrant in any model structure I know. (The spaces are related to configurations spaces of embedded submanifolds).
