Duals of Higher Vector Spaces
Stefano Ronchi, Chenchang Zhu
TL;DR
The paper defines and develops an n-dual for VSn-groupoids by taking internal Hom into the shifted unit B^n R, producing a canonical n-shifted pairing that captures higher-categorical information. It shows this dual is always a VSn-groupoid and is non-degenerate up to homotopy precisely when the original is an n-type, with reflexivity (double dual) in that regime. Using the Dold-Kan correspondence and an internal-Hom version of Eilenberg-Zilber, it places the construction in a robust homotopical framework and analyzes how tensor products and duals interact up to homotopy. The authors compute explicit 1- and 2-duals, connecting the 1-dual to Pradines' dual VB-groupoid and showing the 2-dual is DK-equivalent to the 2-shifted dual in favorable cases, while highlighting limitations and non-degeneracy phenomena beyond low levels. Together, these results provide concrete tools for higher vector spaces and have implications for higher cotangent constructions in derived/differentiable stack contexts and higher symplectic geometry.
Abstract
We introduce a notion of ``$n$-dual'' to a simplicial vector space for $n\ge 0$. Coming with it, there is a canonical 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 in vector spaces, whose $n$-duals are again such $n$-groupoid objects. 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 internal homs. 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 a point.