  Using coinduction and restriction functors, we show that the category of
simplicial $\Bbbk$-vector spaces and the category of $\Bbbk$-representations of
a crossed simplicial group are equivalent after inverting homotopy
equivalences. We do this by first showing that the homotopical version of the
original Dold-Kan equivalence is implemented by an adjoint pair of coinduction
and restriction functors after a suitable homotopy localization. Finally, we
rewrite our equivalence for crossed simplicial objects as a model categorical
equivalence.
