1. Duals of Higher Vector Spaces
- Author
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Ronchi, Stefano and Zhu, Chenchang
- Subjects
Mathematics - Differential Geometry ,Mathematics - Algebraic Topology ,Mathematics - Category Theory ,18D40, 18G31, 18N50, 53D17 - Abstract
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., Comment: 46 pages, preliminary version, comments are welcome!
- Published
- 2024