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

TL;DR

This paper extends the concept of pointwise Kan extensions to 2-categories, providing a new framework using cartesian-marked lax limits and generalizing the Yoneda lemma, with implications for the 2-category of elements.

Contribution

It introduces a novel definition of pointwise Kan extension along discrete 2-opfibrations in a lax 3-category setting, generalizing classical results to higher dimensions.

Findings

Pointwise Kan extensions along discrete 2-opfibrations are always weak.

A new approach using cartesian-marked lax limits is proposed.

Generalization of the parametrized Yoneda lemma to a lax setting.

Abstract

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.