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

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.)

• I do not understand how Definition 3.1 from the paper answers your question --- this definition deals with a completely trivial situation of a (lax) functor from a (classical) category to a (classical) 2-category. Dec 12, 2014 at 17:21
• This is probably because my question is kind of messy even in my head. To put it simple: is there a category of elements for a simplicial/topological/generic V-functor? If not, why? The paper above seems to deal with this situation: $Gr(X)$ is a candidate $\int X$. Dec 12, 2014 at 17:32
• I am still confused. Are you interested in enriched functors $X \rightarrow \mathbb{V}$ or classical functors $C \rightarrow \mathbf{Cat}(\mathbb{V})$? Dec 12, 2014 at 17:56

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': $$X\times_D Y$$.

• The idea behind the comprehension notation comes is the following: existential quantifier = set-comprehension = grothendieck construction. Jan 17, 2015 at 10:02
• Thanks a lot for the explanation! U.P. of Gr(D) apart, I came up precisely with that construction (see also the other discussion mathoverflow.net/questions/191434/… ), so good to know I wasn' wrong. As you say, showing that this is indeed a V-category is lengthy but straightforward: is there a slick proof of the associativity and unitality for the composition map? Jan 18, 2015 at 18:25
• Nah. Not slick - But easy. I'll clean up my notes and post a link to the diagrams sometime next week. Jan 19, 2015 at 15:20