most recent 30 from http://mathoverflow.net 2013-06-20T10:53:43Z http://mathoverflow.net/feeds/question/68470 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/68470/mapping-into-a-geometric-realization Alan Wilder 2011-06-22T02:28:20Z 2011-06-22T02:28:20Z

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