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Suppose $S$ is a simplicial set, $X$ is a space, and we are given a map \[ f: \text{Sing}\,X\to S. \] When is is possible to produce a map $X\to |S|$?

We can take the realization of $f$, to get $|f|:|\text{Sing}\;X|\to |S|$, and then the question becomes: when does $|f|$ factor through the counit map $|\text{Sing}\;X|\to X$ (at least up to homotopy)?

It does if $X$ is a CW-complex, which is not the case for my example. What about if $S$ is a Kan complex? Does that help, or are the properties of $X$ really the key here?

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    $\begingroup$ In the case when $f$ is the identity map, you are sort of hoping for $|f|$ to be a homotopy equivalence, or at least to have a left homotopy inverse. Unlikely if $X$ is not homotopy equivalent to a CW complex. (And note that $|Sing/ X$ is always a Kan complex.) $\endgroup$
    – Tom Goodwillie
    Commented Jun 22, 2011 at 3:57
  • $\begingroup$ Yeah that is more or less what I expected. Thanks, Tom. $\endgroup$
    – Alan Wilder
    Commented Jun 22, 2011 at 6:50

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