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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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Wei Zhan
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  • 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 $f_1$. –  Aug 16 '15 at 07:20
  • @MikeMiller Would you please explain the term 'simplicialization' here? I cannot quite understand it actually. Do we have to triangulate $K\times I$? – Wei Zhan Aug 16 '15 at 07:28
  • 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. –  Aug 16 '15 at 07:30
  • @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? – Wei Zhan Aug 16 '15 at 07:41
  • I mean the latter thing. I'm not really sure I would call it subtle, but to each their own. –  Aug 16 '15 at 13:52

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