A 2-categorical generalization of "final diagram" (I asked for)

Question

Let $H: I\to J$ a $I$-diagram on a category $J$. This is is called $final$ diagram if for any $j\in J$ the comma $j\downarrow F$ is connected (then also non empty). We know the this conciction is equivalent to :

For any functor $F: J \to \mathscr{C} $ the colimit of $F$ exist iff exist the colimit of $F\circ H$ and the canonical morphism $colim(F\circ H) \to colim(F)$ is a isomorphisms.

I ask if is known a generalization to 2-categories and 2-functors or lax functors?

I mean:

For any functor 2-functor $F: J \to \mathscr{C} $ the lax-colimit (or pseudo-colimit) of $F$ exist iff exist the lax-colimit (or pseudo-colimit) of $F\circ H$ and the canonical morphism $colim(F\circ H) \to colim(F)$ is a equivalence.

Thank you in advance.

Answer 1

This paper appears to contain what you asked for.

They use a form of the 2-nerve to give a condition similar to the original one (C/y is connected for all y is the same thing as saying that the nerve of C is "locally contractible" in a suitable sense (see Cisinski 2006)).

(I think that this is the paper where the two authors introduced their so-called 2-nerve).

Answer 2

Lax-pseudo limits, bilimits, or weak-2-limits- choose your terminologoy- don't use non-invertible $2$-cells in their definitions or universal properties. (If you ask for genuine lax or oplax limits, instead of pseudo, then this is different). Hence, you can keep only your invertible $2$-cells, and compute the limits of the associated (2,1)-category (if you have a strict $2$-category, this is just a category enriched in groupoids). In particular, the (2,1)-category will be a an $(\infty,1)$-category in the sense of Lurie. Homotopy limits will BE bilimits in this case.

To go from (2,1)-category as a bicategory to an $\left(\infty,1\right)$-category, first strictify it to a $2$-category- the result is a categeory enriched in groupoids. Apply the nerve functor $N$ Hom-wise to obtain a simplicial category. Now take the homotopy coherent nerve.

If you're given an $\left(\infty,1\right)$-category X, presented as a simplicial set, which is known to be equivalent to a (2,1)-category, you can extract this (2,1)-category explicitly as follows:

Use the left-adjoint to the homotopy-coherent nerve to produce a simplicial category $\mathfrak{C}\left(X\right)$. The hom-simplicial sets will be equivalent to nerves of groupoids. There is a model category struture on the category of groupoids, given by Sharon Hollander, and a pair of adjoint functors

$$\pi_{oid}:sSet \to Gpd$$ $$N:Gpd \to sSet$$ which become a Quillen equivalence between the model-category of groupoids and the left-Bousfield localization of simplicial sets with respect to the map $\partial\Delta^3 \to \Delta^3$ (the so-called $S^2$-nullification). This means that you can simply apply the functor $\pi_{oid}$ Hom-wise to $\mathfrak{C}\left(X\right)$ to obtain a category enriched in groupoids, which is equivalent to the $\left(\infty,1\right)$-category $X$, since $X$ is secretly a (2,1)-category.

This provides the machinery necessary to interpret Lurie's results in terms of actual $2$-categories.