Inner automorphisms as 2-cells
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916934065324032 |
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| author | Hofstra, Pieter Karvonen, Martti |
| author_facet | Hofstra, Pieter Karvonen, Martti |
| contents | Abstract inner automorphisms can be used to promote any category into a 2-category, and we study two-dimensional limits and colimits in the resulting 2-categories. Existing connected colimits and limits in the starting category become two-dimensional colimits and limits under fairly general conditions. Under the same conditions, colimits in the underlying category can be used to build many notable two-dimensional colimits such as coequifiers and coinserters. In contrast, disconnected colimits or genuinely 2-categorical limits such as inserters and equifiers and cotensors cannot exist unless no nontrivial abstract inner automorphisms exist and the resulting 2-category is locally discrete. We also study briefly when an ordinary functor can be extended to a 2-functor between the resulting 2-categories. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_13647 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Inner automorphisms as 2-cells Hofstra, Pieter Karvonen, Martti Category Theory 18A30, 18G45, 18N10 Abstract inner automorphisms can be used to promote any category into a 2-category, and we study two-dimensional limits and colimits in the resulting 2-categories. Existing connected colimits and limits in the starting category become two-dimensional colimits and limits under fairly general conditions. Under the same conditions, colimits in the underlying category can be used to build many notable two-dimensional colimits such as coequifiers and coinserters. In contrast, disconnected colimits or genuinely 2-categorical limits such as inserters and equifiers and cotensors cannot exist unless no nontrivial abstract inner automorphisms exist and the resulting 2-category is locally discrete. We also study briefly when an ordinary functor can be extended to a 2-functor between the resulting 2-categories. |
| title | Inner automorphisms as 2-cells |
| topic | Category Theory 18A30, 18G45, 18N10 |
| url | https://arxiv.org/abs/2406.13647 |