Is there such a thing as a weighted Kan extension?

Question

The title pretty much sums it up.

More in detail. Let $C$, $D$ and $E$ be categories, let $F:C\to D$ and $G:C\to E$ be functors, and let $P:C^{op}\to \mathrm{Set}$ be a presheaf. The colimit of $F$ in $D$ satisfies $$ D(\mathrm{colim} \,F, d) \cong [C,D](F, d) $$ for each object $d$ of $D$, where in the right-hand side $d$ denotes the constant functor, and $[C,D]$ the functor category.

This can be seen as a special case of a Kan extension, which satisfies $$ [E,D](\mathrm{Lan}_G F, K) \cong [C,D](F-,K\circ G-) $$ for each functor $K:E\to D$. Namely, by setting $E$ the terminal category we get exactly a colimit.

Just as well, a colimit is a special case of a weighted colimit, which satisfies $$ D(\mathrm{colim}_W \,C, d) \cong [C^{op}, \mathrm{Set}](W-, D(F-, d)) $$ for each object $d$ of $D$. We get an ordinary colimit by setting $W$ to be the constant presheaf at the singleton.

Now, is there a common generalization?

Note that

• In the Kan extension, the "dependent variable" of $F$ is paired to $K\circ G$, while in the weighted colimit, it is paired to $W$. So it's unclear how to fit both dependencies together.
• One can express Kan extensions as particular weighted colimits - this is not what I'm asking.

(I could ask the same question for the enriched case.)

Any reference would also be welcome.

Answer 1

Yes. Given $F:C\to D$ and a profunctor $H:E$ ⇸ $C$, i.e. a functor $H : C^{\rm op}\times E\to \rm Set$ (or to the enriching category $V$), the $H$-weighted colimit of $F$ is the functor $L : E \to D$ such that each value $L(e)$ is the $W(-,e)$-weighted colimit of $F$ (in a coherent way).

Of course, if $E$ is the unit category this reduces to an ordinary weighted colimit.

On the other hand, if $G:C\to E$ and $H(c,e) = E(G(c),e)$ is the corresponding representable profunctor, this reduces to a (pointwise) Kan extension.

There are real advantages of viewing weighted colimits and Kan extensions in this profunctory light. In particular, this is the natural definition of "weighted (co)limit" that makes sense in the abstract generality of a proarrow equipment or a Yoneda structure. In this paper I found it very useful to obtain a good notion of (co)limit in a new kind of category. It also has good formal properties for relating limits and colimits; see for instance Prop. 8.5 of ibid.