Pointwise Kan extensions along 2-fibrations and the 2-category of elements (Q6589561)

This paper is concerned with the 2-dimensional generalization of the construction of the category of elements [Zbl 0335.18005], which is a natural extension of the usual Grothendieck construction admitting 2-functors from a 2-category \(\mathcal{B}\)\ into \(\mathcal{C}at\), as well as a restriction of the 2-dimensional Grothendieck construction of \textit{I. Bakovic} [Fibrations of bicategories. University of Split, \url{https://www2.irb.hr/korisnici/ibakovic/groth2fib.pdf}] and \textit{M. Buckley} [J. Pure Appl. Algebra 218, No. 6, 1034--1074 (2014; Zbl 1296.18006)]. \textit{M. Lambert} [``Discrete 2-Fibrations'', Preprint, \url{arXiv:2001.11477}] established that discrete 2-opfibrations with small fibers form the essential image of the 2-functor calculating the 2-category of elements. This paper extends this result to 2-equivalences between 2-copresheaves and discrete 2-opfibrations (Theorem 4.14).\N\NIn dimension 1, it is well known that the category of elements can be captured in a more abstract way. Given a copresheaf\N\[\NF:\mathcal{B}\rightarrow\mathcal{S}et\N\]\Nthe construction of the category of elements of \ is equivalently given by the comma object\N\[\N\begin{array} [c]{ccccc} & & \int^{\mathrm{op}}F & & \\\N& \overset{\mathcal{G}\left( F\right) }{\swarrow} & & \searrow & \\\N\mathcal{B} & & \underset{\mathrm{comma}}{\Longleftarrow} & & \boldsymbol{1} \\\N& \underset{F}{\searrow} & & \underset{1}{\swarrow} & \\\N& & \mathcal{S}et & & \end{array}\N\]\Nof which the main theorem of this paper (Theorem 4.11 and Theorem 4.7) is a 2-dimensional generalization. It is shown that an analogous square as the above one exhibits the 2-category of elements \(\mathcal{G}\left( F\right) \) as a lax comma object in 2-\(\mathcal{C}at_{\mathrm{lax}}\)\ and \(F\)\ as the pointwise left Kan extension in 2-\(\mathcal{C}at_{\mathrm{lax}}\)\ of \(\Delta 1\)\ along the discrete 2-opfibration \(\mathcal{G}\left( F\right) \). It is also shown that pointwise Kan extensions in 2-\(\mathcal{C}at_{\mathrm{lax}}\)\ along a discrete 2-opfibration are always weak ones as well, which is established on the basis of an generalization of the parametrized Yoneda lemma, as lax as it can be (Theorem 3.14).\N\NTo give the definition of pointwise Kan extension in 2-\(\mathcal{C}at_{\mathrm{lax}}\), the author takes advantage of the connection between the 2-category of elements and cartesian-marked oplax colimits. Cartesian-marked (op)lax conincal (co)limits area particular case of a 2-dimensional notion of limit introduced by \textit{J. W. Gray} [Formal category theory: Adjointness for 2-categories. Berlin-Heidelberg-New York: Springer-Verlag (1974; Zbl 0285.18006)], which are an alternative to weighted 2-limits. The author [``Colimits in 2-dimensional slices'', Preprint, \url{arXiv:2305.01494}] reduced weighted 2-limits to cartesian-marked lax conical ones, which allows of developing a calculus of colimits in 2-dimensional slices. The author [``2-classifiers via dense generators and Hofmann-Streicher universe in stacks'', Preprint, \url{arXiv:2401.16900}] reduced the study of a 2-classifier to dense generators, constructing a good 2-classifier in stacks and generalizing to dimension 2 the fundamental result that a Grothendieck topos is an elementary topos.