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It's well known that Dold-Kan correspondence is an isomorphism between simplicial vector spaces and non-negative chain complexes of vector spaces. Moreover, weak equivalences and fibrations are preserved.

Could anyone point a reference to not only the statement but also an explicit proof (that preserves weak equivalences and fibrations)? If the proof is simple or straightforward, a hint is appreciated.

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All you ask about is proved in the link you provide or in the references therein. –  Fernando Muro Jun 28 '13 at 14:28
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@FernandoMuro Thanks, could you please be more specific? What I found is "this is discussed for instance in section 4.1 of SchwedeShipley". I have a look at SchwedeShipley, and find the statement without proof or reference, that is why I said "not only the statement but also an explicit proof". –  Ma Ming Jun 28 '13 at 14:38
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Ma Ming, you're right, the proof is not there. Anyway, if you know how the functors are defined, then you'll see that the proof is trivial. –  Fernando Muro Jun 28 '13 at 14:43
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The proof is indeed trivial. There is a subtlety that can trip the unwary: fibrations in "non-negative chain complexes" are not necessarily fibrations in "unbounded chain complexes". In this context, fibrations in non-negative chain complexes are $f:A_\bullet\to B_\bullet$ which are surjective for $\bullet\geq 1$ (but necessarily for $\bullet=0$). –  Charles Rezk Jun 28 '13 at 18:06

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