Duals of Higher Vector Spaces

We introduce a notion of dual n-groupoid to a simplicial vector space for any n 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 VB groupoid over the point.