Geodesic
Pointwise Kan extensions along 2-fibrations and the 2-category of elements
Theory and applications of categories, Tome 41 (2024), pp. 960-994.

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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.
Publié le :
Classification : 18D30, 18A40, 18A30, 18A25, 18N10
Keywords: Grothendieck construction, Kan extension, Grothendieck fibrations, lax comma, 2-categories 
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     author = {Luca Mesiti},
     title = {Pointwise {Kan} extensions along 2-fibrations and the 2-category of elements},
     journal = {Theory and applications of categories},
     pages = {960--994},
     publisher = {mathdoc},
     volume = {41},
     year = {2024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2024_41_a29/}
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Luca Mesiti. Pointwise Kan extensions along 2-fibrations and the 2-category of elements. Theory and applications of categories, Tome 41 (2024), pp. 960-994. http://geodesic.mathdoc.fr/item/TAC_2024_41_a29/