Bicategorical Traces and Cotraces
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866910504918712320 |
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| author | Barhite, Justin |
| author_facet | Barhite, Justin |
| contents | The familiar trace of a square matrix generalizes to a trace of an endomorphism of a dualizable object in a symmetric monoidal category. To extend these ideas to other settings, such as modules over non-commutative rings, the trace can be generalized to a bicategory equipped with additional structure called a shadow. We propose a notion of bicategorical cotrace of certain maps involving dualizable objects in a closed bicategory equipped with a coshadow, and we use this framework to draw connections to work of Lipman on residues and traces with Hochschild (co)homology, and to work of Ganter and Kapranov on 2-representations and 2-characters. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_13070 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Bicategorical Traces and Cotraces Barhite, Justin Category Theory 16D90, 18D15, 18M05, 18M30, 18N10 (Primary) 13D03, 16D20, 16E40, 20C15 (Secondary) The familiar trace of a square matrix generalizes to a trace of an endomorphism of a dualizable object in a symmetric monoidal category. To extend these ideas to other settings, such as modules over non-commutative rings, the trace can be generalized to a bicategory equipped with additional structure called a shadow. We propose a notion of bicategorical cotrace of certain maps involving dualizable objects in a closed bicategory equipped with a coshadow, and we use this framework to draw connections to work of Lipman on residues and traces with Hochschild (co)homology, and to work of Ganter and Kapranov on 2-representations and 2-characters. |
| title | Bicategorical Traces and Cotraces |
| topic | Category Theory 16D90, 18D15, 18M05, 18M30, 18N10 (Primary) 13D03, 16D20, 16E40, 20C15 (Secondary) |
| url | https://arxiv.org/abs/2307.13070 |