Mike Shulman: This is an interesting notion, but it is not a special case of the general notion of weighted limit. The weight for a weighted limit of a diagram $F\colon D\to C$ is a functor $W\colon D\to Set$, not $C\to Set$. Since $D$ is the empty category for initial objects, no nontrivial weights are possible in that case. Of course, what you call a “$W$-weighted initial object” is the same a representation of $W$.

Domenico Fiorenza: I see the choice $C$ for the category was a bad one.. I had written everithing in terms of $C$ here since when there’s just one category in sight it’s customary to call it $C$, but if one is reading at this entry with weighted limits in mind, this should rather be $D$. Now I’ll edit the page using $D$ instead; I hope this will make clear what I had in mind.

Mike Shulman: Changing the name of the category doesn’t change the meaning of the definition, does it? I don’t understand how this is supposed to make it clearer.

Domenico Fiorenza: My fault: when you said that the weight for a weighted limit of a diagram $F\colon D\to C$ is a functor $W\colon D\to Set$, not $C\to Set$, I guessed that in the way I had written things I had made you get confused, since in the previous version of this page $W$ was a functor from $C$ to $Set$. Thus I tried to clarify thing changing the notation accordingly.

So let me write here what I am suggesting the weighted limit $lim_WF$ could be, for a functor $F:D\to C$, to see if I’m able to make the aim of this entry clearer: $lim_WF$ should be a terminal object in the category of functors from $\{\top\}\star_WD$ to $C$ exteding $F$; here $\{\top\}\star_WD$ is the category obtatined by adding a $W$-weighted initial object to $D$.

Mike Shulman: Ah, yes! It is quite right that a $W$-weighted cone (or cylinder) over a diagram $F\colon D\to C$ is the same as a functor from the cograph of $W$ (considered as a profunctor $D ⇸ 1$) to $C$ which restricts to $F$ along the inclusion of $D$. In fact, this is the basis for a natural generalization of weighted limits, where we consider profunctors $D ⇸ K$ for arbitrary $K$, rather than merely $K=1$; there are some remarks about this generalization at proarrow equipment.

I would suggest, however, that you not use the phrase “$W$-weighted initial object” since that construction isn’t actually itself a weighted (co)limit; it’s just something you can use to help define the weighted limit.

Mike Shulman: It seems as though possibly you would want to start from a functor $W\colon K^{op}\times D\to Spaces$, rather than simply $D\to Set$. In that case, perhaps your “weighted join” is the same as the cograph of a profunctor (in the quasicategory context)?

Fosco: I think Mike Shulman is right, as I was motivated to open this question on MO in order to clarify the role of profunctors in the definition of limits. It seems pretty obvious to me that the generalization which leads to consider representing object $\lim \; F$ for the functor $M^\text{op}\to \Sets$ sending $m\in M$ to $\Cones (m,F)\cong \Sets^C(*, M(m,F))$ can be easily carried on to consider

• representing object $\lim^W \; F$ for a general functor $W\colon C\to \Sets$ and the functor $m\mapsto \Sets^C(W, M(m,F))$;
• representing object $\lim^W \; F$ for a general functor $W\colon C\to V$ and the functor $m\mapsto [C,V](W, M(m,F))$, for $V$ a suitable Benabou cosmos?.

This led me to wonder if, in the same vein that regards the good old limit $\lim \; F$ as the terminal object in the category $\text{Fun}^F([0]\star C,D)$, where $[0]\star C$ is the collage along the terminal profunctor, then the limit of $F$ weighted by $W\colon C\to \Sets$ can be seen as the terminal object in the category $\text{Fun}^F([0]\star_W C,D)$, where $[0]\star_W C$ is merely the cograph of $W$ (I’m certain you see the pattern!).