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Dold-Kan correspondence
The two categories (positively graded chain complexes over AbGp and simplicial AbGps) are equivalent.
Relation to Hurewicz: 
Dold-Kan correspondence
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2021-10-03
Alternatively: take subcategory of fibrant objects, observe enrichment over chain complexes, apply Dold-Kan to get a simplicial enrichment, then take the homotopy coherent nerve or simplicial nerve.
Getting a chain complex from a simplicial set : take free \({\mathbf{Z}}{\hbox{-}}\)modules levelwise, then apply Dold-Kan.