# A 2-categorical generalization of “final diagram” (I asked for)

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.

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The property of being cofinal (what you're calling final) is at least covered for oo-categories in Lurie's HTT chapter 4.1. The original proof of this fact for homotopy colimits in the model category case is due to Quillen and called Quillen's theorem A. If your 2-category happens to be groupoidal above the 1-morphisms though, this generalization should do the trick. – Harry Gindi Nov 12 '10 at 21:36

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).

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Hi Harry. I saw the Lurie work, and Bullejos, articles, I seems that they are about homotopical limits. But anyway thank you – Buschi Sergio Nov 13 '10 at 16:03
@Buschi: I would have guessed that you could specialize the homotopical version to the ordinary version by using some kind of trivial homotopical structure. – Harry Gindi Nov 13 '10 at 16:15
@Gindi. Thank you anyway for your interesting help. But I mean a stright genaralization to lax-Pseudo funcors, now I have edit i detail what I mean. – Buschi Sergio Nov 13 '10 at 18:12

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.

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Thank you very much, I'll read the great and hard work of Lurie, I know almost nothing about homotopy limits, but are very interesting. – Buschi Sergio Nov 13 '10 at 21:17
Reading Lurie on your own can be quiet daunting. N-lab may be a helpful companion :). Anyway, for such a question you don't need the full machinery of Lurie, and it might be easier to just work out on your own by analogy from $1$-category theory. However, once you learn the yoga of infinity categories, it can become quite useful. – David Carchedi Nov 13 '10 at 21:50