PB-groupoids vs VB-groupoids
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912916447428608 |
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| author | Garmendia, Alfonso Cattafi, Francesco |
| author_facet | Garmendia, Alfonso Cattafi, Francesco |
| contents | In this paper we establish the principal bundle counterpart of the well-known and widely applied notion of vector bundle groupoid (VB-groupoid). In particular, we provide a general notion of principal bundle groupoid (PB-groupoid) as a diagram of Lie groupoids and principal bundles, together with the action of a (strict) Lie 2-groupoid. This recovers as particular cases various constructions involving structural Lie 2-groups, Lie groupoids and Lie groups. Moreover, given any VB-groupoid, we detect which frames are compatible with the underlying structure and show that the set of such frames has a natural PB-groupoid structure, with the action of the general linear 2-groupoid of the appropriate ranks. We use this construction, as well as that of the associated bundle induced by a 2-representation, to extend the classical correspondence between vector bundles and principal bundles to a new correspondence between VB-groupoids and PB-groupoids. We conclude with several examples and applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_06259 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | PB-groupoids vs VB-groupoids Garmendia, Alfonso Cattafi, Francesco Differential Geometry In this paper we establish the principal bundle counterpart of the well-known and widely applied notion of vector bundle groupoid (VB-groupoid). In particular, we provide a general notion of principal bundle groupoid (PB-groupoid) as a diagram of Lie groupoids and principal bundles, together with the action of a (strict) Lie 2-groupoid. This recovers as particular cases various constructions involving structural Lie 2-groups, Lie groupoids and Lie groups. Moreover, given any VB-groupoid, we detect which frames are compatible with the underlying structure and show that the set of such frames has a natural PB-groupoid structure, with the action of the general linear 2-groupoid of the appropriate ranks. We use this construction, as well as that of the associated bundle induced by a 2-representation, to extend the classical correspondence between vector bundles and principal bundles to a new correspondence between VB-groupoids and PB-groupoids. We conclude with several examples and applications. |
| title | PB-groupoids vs VB-groupoids |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2406.06259 |