  We introduce a notion of dual $n$-groupoid to a simplicial vector space for
any $n\ge 0$. This has a canonical duality pairing which we show to be
non-degenerate up to homotopy for homotopy $n$-types. As a result this notion
of duality is reflexive up to homotopy for $n$-types. In particular the same
properties hold for $n$-groupoid objects, whose $n$-duals are again
$n$-groupoids. We study this construction in the context of the Dold-Kan
correspondence and we reformulate the Eilenberg-Zilber theorem, which
classically controls monoidality of the Dold-Kan functors, in terms of mapping
complexes. We compute explicitly the 1-dual of a groupoid object and the 2-dual
of a 2-groupoid object in the category of vector spaces. As the 1-dual of a
groupoid object we recover its dual as a $\mathsf{VB}$ groupoid over the point.
