Duals of Higher Vector Spaces

TL;DR

This paper develops a theory of duals for higher vector spaces and groupoid objects, establishing non-degenerate pairings up to homotopy and exploring their properties within the Dold-Kan correspondence.

Contribution

It introduces the concept of $n$-duals for simplicial vector spaces and $n$-groupoid objects, proving their reflexivity up to homotopy and connecting to classical theorems like Eilenberg-Zilber.

Findings

Non-degenerate pairings up to homotopy for $n$-types.

Explicit computation of 1-dual and 2-dual objects.

Reformulation of the Eilenberg-Zilber theorem in terms of internal homs.

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.