# Is there such a thing as a weighted Kan extension?

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.

• Have you seen Chapter 4 in Kelly's book on enriched categories? Feb 28 '20 at 22:34

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.