This is a crosspost of [this MSE question](https://math.stackexchange.com/questions/1433939/the-category-of-elements-enrichment-and-weighted-limits).


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Every so often, when reading notes online or skimming through books, the [category of elements](https://en.wikipedia.org/wiki/Category_of_elements) and the [Grothendieck construction](https://en.wikipedia.org/wiki/Grothendieck_construction) pop up. I don't know anything about the Grothendieck construction, and I don't understand the significance of its special case - the category of elements. This is probably going to be the first in a series of questions about the Grothendieck construction (I'll interlink with later ones if/when I ask them).

My main motivation for asking this question is the excerpt below from Borceux vol II, sec 6.6. I have tried to go into detail about how I don't understand the significance of the category of elements in this context.

> The first observation for enriching the notion of limit is the fact in many fundamental results of the previous chapters, we had to deal with a situation like in diagram 6.3.6, where $F,G$ are functors and $\mathsf{Elts}(G)$ is the category of elements of $G$; the limit considered was that of $F\circ \phi _G$ [...]. Restricting one's attention to limits of the form $\varprojlim (F\circ \phi _G)$ is not at all a restriction since, choosing for $G$ the constant functor on the singleton, the category of elements of $G$ is just $\mathscr A$ itself and $\phi _G =1_\mathscr{A}$, so that we recapture the limit of $F$.
>
>[![enter image description here][1]][1]
>
>But the key observation for enriching the notion of limit is the following fact
>
>**Lemma 6.6.1** In the situation which has just been described, there exist bijections natural in $B\in \mathscr B$,
$$\mathsf{Nat}(\varDelta_B,F\circ \phi _G)\cong \mathsf{Nat}(G,\mathscr B(B,F-))$$ where $\varDelta _B:\mathsf{Elts}(G)\longrightarrow \mathscr B$ is the constant functor on the object $B\in \mathscr B$.

From this lemma Borceux concludes that $\varprojlim (F\circ \phi_G)$ exists iff $\exists L\in \mathscr B$ with bijections natural in $B$:
$$\mathsf{Nat}(G,\mathscr B(B,F-))\cong \mathscr B(B,L)$$
And from this, he almost *has* to write the definition of a weighted limit.


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Alternatively, here's an approach I find somewhat more intuitive, but one that does not explicitly mention the category of elements at all.
In classical CT, we define limits using $\varDelta$, which is not available in a general enriched category $\mathsf C$ since $\mathsf C(A,B)$ has no elements to work with. In the unenriched case, one could also try to define $\varDelta$ via the cartesian closed structure of $\mathsf{Cat}$ as a right adjoint of the projection $\pi:\mathsf C\times \mathsf J\longrightarrow \mathsf C$, but this doesn't generalize either since most enriched categories are not cartesian closed. So we're left with trying the representability approach - a limit of $F$ is just a representation of $\mathsf{Nat}(\varDelta-,F)$. Here we can escape the "problems of enrichment" because I *think* the following holds:

**Proposition** There's a bijection natural in $X$: $\mathsf{Nat}(\varDelta \ast,\mathsf{Hom}(X,F-))\cong \mathsf{Nat}(\varDelta X,F)$, where $\ast$ is the singleton.

The point is to now only mention the functor $\varDelta \ast$ instead of the general $\varDelta$.

*Digression* - is the above bijection also natural in $F$? (I think it is.)

Now since $\varprojlim F$ is a representation of $\mathsf{Nat}(\varDelta -,F)$, we can *define* it as an object with bijections natural in $X$$$\mathsf{Hom}(X,\varprojlim F) \cong \mathsf{Nat}(\varDelta \ast,\mathsf{Hom}(X,F-))$$

Now we have reduced the usage of $\varDelta$ to the functor $\varDelta \ast$ alone. The role of $\varDelta \ast$ as a weight is also easily seen, as each component of natural transformation $\eta _A :\varDelta \ast \Rightarrow \mathsf{Hom}(X,F-)$ picks out an arrow $X\rightarrow FA$, and so replacing 
$\varDelta \ast$ by say, $\varDelta (\ast\coprod \ast)$ would give a "double cone". Replacing $\varDelta \ast$ by more general functors $G$ would produce more intricate cones.

This alternative approach is (I think) completely ignorant of the category of elements, and yet manages to give a nice motivation for the definition of a weighted limit. On the other hand, Borceux's approach just poops it out by category-of-elements-magic.

> What exactly is happening here? What is the role of the category of elements in this story and why does it effortlessly yield the definition of a weighted limit?

  [1]: https://i.sstatic.net/NELt5.png