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I have been given the following definition of simplicial approximation in lectures:

Let $K, L$ be simplicial complexes and $f : |K| \to |L|$ be a continuous map of their polyhedra. A simplicial approximation of $f$ is a map $g$ of vertices of $K$ to vertices of $L$ such that $$f(\mathrm{st}_K(v)) \subseteq \mathrm{st}_L(g(v))$$ for each vertex $v$ in $K$.

However, elsewhere I find the definition that a simplicial approximation is any simplicial map which is homotopic to the original map. This seems, to me, to be considerably more general than the definition I have, since, for example, if $f$ is homotopic to a constant map, then I can have very trivial simplicial approximations in this sense. So my question is, which is the more common / "morally" correct / better definition?

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  • $\begingroup$ I think we're just using this definition to help organize proofs. It's nice to have an easy-to-check criterion to prove things about. $\endgroup$ – Qiaochu Yuan Feb 25 '11 at 20:19
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    $\begingroup$ What one usually wants is simply to replace your functions with simplical maps homotopic to them. On the other hand, one eventually ends up needing the more stringent condition you mention, or something like it, because it is awfully convenient. $\endgroup$ – Mariano Suárez-Álvarez Feb 25 '11 at 20:31
  • $\begingroup$ @Mariano: I think that's the answer I was looking for, although it would be nice to see some examples of results where the use of simplicial approximations in the stronger sense is essential. If you could post an elaboration of your comment as an answer I'll accept it. $\endgroup$ – Zhen Lin Feb 27 '11 at 16:36
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Mariano's comment is dead on. It might be helpful to think also of cellular approximation. Given a map of CW complexes $f: X \to Y$, $g: X \to Y$ is a cellular approximation of $f$ if they are homotopic and $g$ is cellular, that is it restricts to a map between the n-skeletons of $X$ and $Y$. I personally don't think a lot about simplicial approximations or cellular approximations because I know they always exist in the settings I care about. The point of having these approximations is then you can make standard types of arguments (like inductive arguments) that are more familiar without losing any generality.

An interesting question is whether or not cellular/simplicial maps are cofibrant objects in a particular/obvious model category structure on the category of morphisms (if such a model structure exists). Maybe I should ask such a question on some sort of math question site...

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  • $\begingroup$ Wait, the category of CW complexes does not have colimits, right? So what would it mean to have a model structure on the category of cellular/simplicial maps? $\endgroup$ – Akhil Mathew Feb 26 '11 at 2:36
  • $\begingroup$ I meant that if you look at the category of where objects are maps of spaces/sSets/CGH/whatev and morphisms are squares are the simplicial/cellular maps cofibrant. $\endgroup$ – Sean Tilson Feb 26 '11 at 4:07
  • $\begingroup$ NM, I see your point... there must be some way to make sense out of it... $\endgroup$ – Sean Tilson Feb 26 '11 at 4:37

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