Pontrjagin duality on multiplicative Gerbes
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2020
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| _version_ | 1866914965957378048 |
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| author | Blanco, Jaider Uribe, Bernardo Waldorf, Konrad |
| author_facet | Blanco, Jaider Uribe, Bernardo Waldorf, Konrad |
| contents | We use Segal-Mitchison's cohomology of topological groups to define a convenient model for topological gerbes. We introduce multiplicative gerbes over topological groups in this setup and we define its representations. For a specific choice of representation, we construct its category of endomorphisms and we show that it induces a new multiplicative gerbe over another topological group. This new induced group is fibrewise Pontrjagin dual to the original one and therefore we called the pair of multiplicative gerbes `Pontrjagin dual'. We show that Pontrjagin dual multipliciative gerbes have equivalent categories of representations and moreover, we show that their monoidal centers are equivalent. Examples of Pontrjagin dual multiplicative gerbes over finite and discrete, as well as compact and non-compact Lie groups are provided. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_05056 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Pontrjagin duality on multiplicative Gerbes Blanco, Jaider Uribe, Bernardo Waldorf, Konrad Algebraic Topology Category Theory 53C08, 55U30 (primary), 55N30 (secondary) We use Segal-Mitchison's cohomology of topological groups to define a convenient model for topological gerbes. We introduce multiplicative gerbes over topological groups in this setup and we define its representations. For a specific choice of representation, we construct its category of endomorphisms and we show that it induces a new multiplicative gerbe over another topological group. This new induced group is fibrewise Pontrjagin dual to the original one and therefore we called the pair of multiplicative gerbes `Pontrjagin dual'. We show that Pontrjagin dual multipliciative gerbes have equivalent categories of representations and moreover, we show that their monoidal centers are equivalent. Examples of Pontrjagin dual multiplicative gerbes over finite and discrete, as well as compact and non-compact Lie groups are provided. |
| title | Pontrjagin duality on multiplicative Gerbes |
| topic | Algebraic Topology Category Theory 53C08, 55U30 (primary), 55N30 (secondary) |
| url | https://arxiv.org/abs/2012.05056 |