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I know Dold-Kan actually holds for any abelian category, i.e gives an equivalence between simplicial objects in a fixed abelian category and the connective chain complexes over it.

I've never seen an actual proof of this fact though, so I was wondering whether in the case of a general symmetric monoidal abelian category, the Alexander-Whitney and shuffle maps still exist and give a homotopy equivalence between the transferred pointwise tensor product of complexes and the usual one making the category of chain complexes of say abelian groups symmetric monoidal closed.

Are the details written anywhere?

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§1.2.3 in Lurie's Higher Algebra establishes the Dold—Kan correspondence for idempotent complete additive categories and constructs the Alexander—Whitney maps in this generality.

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