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Let $X$ be a paracompact topological space and choose a good cover $U_i$ of $X$. Remember that a good cover is one that consists of open subsets, such that each set $U_i$ is contractible and all finite intersections $U_{i_1} \cap U_{i_2} \cap \dots \cap U_{i_k}$ are either empty or contractible.

This induces a simplicial topological space $U_{\bullet}$. Let $Z_{\bullet}$ be another simplicial topological space. I keep reading that any continuous map $$ f \colon |U_{\bullet}| \to |Z_{\bullet}| $$ is homotopic to a simplicial one (e.g. in the sketched proof of theorem 4.5 here). Is there any reference for this? How do I prove this?

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  • $\begingroup$ This is an easy corollary of the well-known simplicial approximation theorem. $\endgroup$
    – Fernando Muro
    Commented Mar 19, 2012 at 20:54
  • $\begingroup$ But isn't the simplicial approximation theorem (at least as I know it) a statement about simplicial complexes and not about simplicial spaces? See also <a href="mathoverflow.net/questions/90480/…> question. $\endgroup$
    – Ulrich Pennig
    Commented Mar 19, 2012 at 21:04
  • $\begingroup$ sorry for the broken link. I meant: mathoverflow.net/questions/90480/… $\endgroup$
    – Ulrich Pennig
    Commented Mar 19, 2012 at 21:44
  • $\begingroup$ The property stated above is not correct (as suggests the link). What is true is that, there exists another good cover $U'$ of $X$ and a morphism of simplicial spaces $U'_\bullet\to Z_\bullet$ whose realization is homotopic to $f$ under the weak homotopy equivalences $|U_\bullet|\sim X\sim |U'_\bullet|$. $\endgroup$
    – D.-C. Cisinski
    Commented Mar 19, 2012 at 23:12
  • $\begingroup$ @Ulrich: Sorry, I took simplicial complex for simplicial space. $\endgroup$
    – Fernando Muro
    Commented Mar 20, 2012 at 11:28

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