Pointwise Kan extensions along 2-fibrations and the 2-category of elements
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917751137763328 |
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| author | Mesiti, Luca |
| author_facet | Mesiti, Luca |
| contents | We study the 2-category of elements from an abstract point of view. We generalize to dimension 2 the well-known result that the category of elements can be captured by a comma object that also exhibits a pointwise left Kan extension. For this, we propose an original definition of pointwise Kan extension along a discrete 2-opfibration in the lax 3-category of 2-categories, 2-functors, lax natural transformations and modifications. Such definition uses cartesian-marked lax limits, which are an alternative to weighted 2-limits. We show that a pointwise Kan extension along a discrete 2-opfibration is always a weak one as well. The proof is based on an original generalization of the parametrized Yoneda lemma which is as lax as it can be. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_04566 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Pointwise Kan extensions along 2-fibrations and the 2-category of elements Mesiti, Luca Category Theory Logic 18D30, 18A40, 18A30, 18A25, 18N10 We study the 2-category of elements from an abstract point of view. We generalize to dimension 2 the well-known result that the category of elements can be captured by a comma object that also exhibits a pointwise left Kan extension. For this, we propose an original definition of pointwise Kan extension along a discrete 2-opfibration in the lax 3-category of 2-categories, 2-functors, lax natural transformations and modifications. Such definition uses cartesian-marked lax limits, which are an alternative to weighted 2-limits. We show that a pointwise Kan extension along a discrete 2-opfibration is always a weak one as well. The proof is based on an original generalization of the parametrized Yoneda lemma which is as lax as it can be. |
| title | Pointwise Kan extensions along 2-fibrations and the 2-category of elements |
| topic | Category Theory Logic 18D30, 18A40, 18A30, 18A25, 18N10 |
| url | https://arxiv.org/abs/2302.04566 |