# Is the counit of geometric realization a Serre fibration?

Recall that a Serre fibration between topological spaces is a map which has the homotopy lifting property (HLP) for all CW complexes (equivalently for all disks $D^k$). The Serre fibrations are the fibrations in a model category structure on topological spaces in which the weak equivalences are the weak homotopy equivalences. The cofibrant objects include the CW-complexes.

The geometric realization and singular simplicial set functors from an adjunction $$|-|: sSet \leftrightarrows Top: Sing_*$$ which is in fact a Quillen equivalence between the above model structure and the standard model structure on simplicial sets.

The geometric realization of every simplicial set is a CW-complex and the counit map $$\epsilon_X: |Sing(X)| \to X$$ is a weak homotopy equivalence. Hence this gives a nice functorial CW-approximation for every space. Moreover $|Sing(X)|$, being a CW-complex, is cofibrant.

I would like this to be a cofibrant replacement of $X$, however for that I should further require that the counit map $\epsilon_X$ is also a Serre fibration. This is not so obvious to me because to check the lifting property I have to map into a geometric realization (which is a bit subtle). However using cellular approximation I have been able to show that this map has a weaker fibration property; it is a sort of "Serre fibration version" of a Dold fibration (where we only have a weak homotopy lifting property). This is good enough for many applications, but it still leads me to ask my question:

Is the counit map $\epsilon_X$ a Serre fibration? Are there conditions on X (such as paracompactness) which will make this hold true? If it is not a Serre fibration generally, what is the easiest counterexample?

Another related question is whether in Top there is a functorial cofibrant replacement by CW-complexes. The usual approach would probably be to use the small object argument, which would result in a cellular space (a space like a CW-complex but with the cells possibly attached out of order). I suspect that a little clever handicrafting can make this yield an actual CW-complex. Is that intuition correct?

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+1, but a metaphorical +10 for using Dold-Serre fibrations, (which, incidentally, have a long-exact sequence in homotopy) which I've never really seen discussed. As for the last question, perhaps this upcoming talk at CT2013 will be of interest to you: web.science.mq.edu.au/groups/coact/seminar/ct2013/abstracts/… –  David Roberts Jun 24 '13 at 12:29