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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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    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.2011-02-25
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    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.2011-02-25
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    @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.2011-02-27

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