Duals of Higher Vector Spaces
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909929213788160 |
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| author | Ronchi, Stefano Zhu, Chenchang |
| author_facet | Ronchi, Stefano Zhu, Chenchang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_03306 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Duals of Higher Vector Spaces Ronchi, Stefano Zhu, Chenchang Differential Geometry Algebraic Topology Category Theory 18D40, 18G31, 18N50, 53D17 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. |
| title | Duals of Higher Vector Spaces |
| topic | Differential Geometry Algebraic Topology Category Theory 18D40, 18G31, 18N50, 53D17 |
| url | https://arxiv.org/abs/2407.03306 |