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In the wikipedia page simplicial approximation theorem and a former answer related to this theorem, it was mentioned that on simplicial complexes, homotopy between continuous mappings can be approximated using simplicial mappings and subdivisions.

Anyone knows the exact statement of this theorem? Thanks in advance.

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  • $\begingroup$ Given a map $f: K \to L$ between simplicial complexes, $f$ can be homotoped by an arbitrarily small amount to obtain a simplicial map between subdivisions of $K$ and $L$. Given a homotopy $f_t: K \times I \to L$, the same is true; if $f_0$ and $f_1$ are both already simplicial maps, then one may choose the 'simplicialization' to not change $f_0$ or $f_1$. $\endgroup$
    – user98602
    Aug 16, 2015 at 7:20
  • $\begingroup$ @MikeMiller Would you please explain the term 'simplicialization' here? I cannot quite understand it actually. Do we have to triangulate $K\times I$? $\endgroup$
    – Wei Zhan
    Aug 16, 2015 at 7:28
  • $\begingroup$ I just made it up. It means the map we picked, homotopic to the original, that is now a simplicial map. Yes, if $K$ is a simplicial complex, $K \times I$ naturally inherits the structure of one. $\endgroup$
    – user98602
    Aug 16, 2015 at 7:30
  • $\begingroup$ @MikeMiller The problem I met here might just be the 'inheritance'. Do we view $K\times I$ as the categoric product, or as a subtle one described here which is truly a triangulation? $\endgroup$
    – Wei Zhan
    Aug 16, 2015 at 7:41
  • $\begingroup$ I mean the latter thing. I'm not really sure I would call it subtle, but to each their own. $\endgroup$
    – user98602
    Aug 16, 2015 at 13:52

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