The (un)reasonable (non-)ubiquity of the Grothendieck construction

Question

Is there a way to export the Grothendieck construction to different contexts than $Cat$? in theory, if you build $\int F$ out of $F\colon \mathcal C\to Cat$, or $F\colon \mathcal C\to Sets$, as a suitable (lax) pullback along the canonical forgetful $Cat_*\to Cat$, $Sets_*\to Sets$, you can do the same replacing $Sets_*$ with pointed categories, pointed spaces, pointed simplicial sets and their unpointed analogues.

The central importance of the category of elements in both Category Theory and Geometry makes me wonder if there is a well-established, abstract way to perform these constructions, capturing the various examples. Is it possible to talk about a Grothendieck construction for suitable functors $F\colon \mathcal C\to $ "something", proveided this something is "nice"?

More precisely, is there a "general Grothendieck construction in enriched category theory"? If not (I firmly believe the answer is no, given that weighted limits are seldom reducible to conical ones, and more importantly in non-cartesian situations "conical limits" make no sense), how can I enucleate a set of properties on $\cal V$ in such a way that there is a Grothendieck construction for $\cal V$-presheaves?

Edit: apparently the answer to the previous questions is yes. But now the validity of this machinery (Definition 3.1 and the rest of section 3) makes me wonder if the following result remains true in enriched setting.

The following properties for a functor $F\colon \mathcal C\to \bf Sets$ are equivalent:

1. $F$ commutes with finite limits;
2. The Yoneda extension $\text{Lan}_{y}F$ commutes with finite limits;
3. $F$ is a filtered colimit of representable functors;
4. The category of elements $\int F$ of $F$ is cofiltered.

If yes, how's it proved? (Haven't thought about it yet, so the answer could be trivial.)

Answer 1

Given a $\mathbb V$-functor $F:X^\mathrm{op}\to \mathbb V$, I came up with the following answer some years back:

Consider a $\mathbb V$-distributor $D:X^\mathrm{op}\otimes Y \to \mathbb V$. The Grothendieck construction $\mathrm{Gr}(D)$ is then the following $\mathbb V$-category.

Objects are triples $\phi=(x,\phi,y)$ with $x\in X$, $y\in Y$ and $\phi:\mathbf 1\to D(x,y)$ a morphism in $\mathbb V$.

Morphism objects are defined the following way. Given $\phi,\psi\in\mathrm{Gr}(D)$ we define $[\phi_1,\phi_2]$ to be the pullback

$$[x_1,x_2]\times_{D(x_1,y_2)}[y_1,y_2].$$

As an alternative notation I found a notation remeniscent of set-comprehension usefull:

$$\mathrm{Gr}(D)=:\{ x\in X \,\rangle\, D(x,y) \,\langle\, y\in Y \}$$

Showing that this is indeed a $\mathbb V$-category is lengthy but straightforward. The constrution exhibits the following universal property:

First, we define a $\mathbb V\!-\!\mathrm{Cat}$-category of $\mathbb V$-distributors.

Objects are triples $(X,D,Y)$ with $D:X^\mathrm{op}\otimes Y\to \mathbb V$.

Morphism $\mathbb V$-categories are defined as

$$\{ (L,R)\in [X_1,X_2]\otimes [Y_1,Y_2] \,\rangle\, [D_1(-,-),D_2(L-,R-)] \,\langle\, \mathbb{1}_{\mathbb V-\mathrm{Cat}} \}.$$

Denoting this category $\mathbb V-\mathrm{Dist}$ we find a $\mathbb V-\mathrm{Cat}$-equivalence (constituting a $\mathbb V-\mathrm{Cat}$-adjunction)

$$\mathbb V-\mathrm{Cat}[Z,\mathrm{Gr}(D)]\simeq \mathbb V-\mathrm{Dist}(\mathrm{hom}(Z),D)$$

where $\mathrm {hom}$ is the functor assigning to a $\mathbb V$-category its $\mathrm {hom}$-distributor. In this sense one could call $\mathrm{Gr}(D)$ the covering category of $D$. This adjunction is a straightforward generalisation of the one given in Thomas Streichers notes "Distributors à la Jean Benabou". Another useful name/notation might be 'fibre product': $X\times_D Y$.