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Pointwise Kan extensions along 2-fibrations and the 2-category of elements

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Luca Mesiti

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.

Keywords:
Grothendieck construction, Kan extension, Grothendieck fibrations, lax comma, 2-categories

2020 MSC:
18D30, 18A40, 18A30, 18A25, 18N10

*Theory and Applications of Categories,*
Vol. 41, 2024,
No. 30, pp 960-994.

Published 2024-08-13.

http://www.tac.mta.ca/tac/volumes/41/30/41-30.pdf

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