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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).

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2 Answers 2

up vote 10 down vote accepted

To get such a result we typically need that the degenerate subspaces include via cofibrations, and we can get a counterexample by picking a standard non-cofibration.

Let $X_0 = \{0\}$, and let $X_1 = \{0, 1, 1/2, 1/3, \ldots\} \subset \Bbb R$, with degeneracy $s^0: X_0 \to X_1$ being the natural inclusion. Build the rest of $X$ so that all higher simplices are degenerate: let $X_p = X_1 \vee \cdots \vee X_1$ with $p$ factors. The face maps are collapsing down a factor on the side, or using the fold map on two adjacent factors; the degeneracies are obtained by inserting a $0$.

Let $Y$ be the simplicial space with the same underlying set, but where we've now given everything the discrete topology. The natural map $Y \to X$ is a levelwise equivalence because both are totally disconnected.

Both simplicial spaces have only a single $0$-simplex, some space of $1$-simplices, and all higher simplices are degenerate. The geometric realization of such a $Z$ is the space $$ \frac{Z_1 \times [0,1]}{Z_1 \times \{0,1\} \cup \{*\} \times [0,1]} = Z \wedge S^1. $$ Thus, the map $|Y| \to |X|$ is the standard continuous bijection from a countable wedge of circles to $\{0,1,1/2,1/3,\ldots\} \wedge S^1$, which is homeomorphic to the Hawaiian earring (via the standard "continuous bijection from compact to Hausdorff" argument).

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Let me motivate this construction by isolating a counterexample to a different property. $N\to X_1$ is a map that is a weak equivalence, but that ceases to be a weak equivalence on taking suspension, because $\pi_0$ doesn't care about points getting close, but $\pi_1$ does. Now we want to promote this to a geometric realization, which is plausible since suspension is related to a homotopy colimit. –  Ben Wieland Jun 11 at 4:17
    
Yes, this makes me wonder if the result holds if we restrict to $\Delta$-generated spaces? Then the example goes away, but maybe there is a different counter example in that case? –  Chris Schommer-Pries Jun 11 at 15:40

I think this can be treated using a gluing theorem for homotopy equivalences which is discussed in my answer to this stackexchange question. You have to represent the realisation of a simplicial space as a repeated adjunction space, but that is not too hard.

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