## Recall: Limits

Recall that the **limit** of a functor $D\colon\mathcal{I}\to\mathcal{C}$ is, if it exists, the pair $(\mathrm{lim}(D),\pi)$ with

- $\lim(D)$ an object of $\mathcal{C}$, and
- $\pi\colon\Delta_{\lim(D)}\Rightarrow D$ a cone of $\lim(D)$ over $D$

such that the natural transformation $$\pi_*\colon h_{\lim(D)}\Rightarrow\mathrm{Cones}_{(-)}(D),$$ is a natural isomorphism, where

- $\mathrm{Cones}_{(-)}(D)\overset{\mathrm{def}}{=}\mathrm{Nat}(\Delta_{(-)},D)$, and
- The component at $X\in\mathrm{Obj}(\mathcal{C})$ of $\pi_*$ is the map $(\pi_*)_X \colon \mathrm{Hom}_\mathcal{C}(X,\lim(D))\to \mathrm{Cones}_X(D)$ sending a morphism $f\colon X\to\lim(D)$ to the cone $$\Delta_X\xrightarrow{\Delta_f}\Delta_{\lim(D)}\to D$$ of $X$ over $D$.

## Recall: Ends

Now, the **end** of a functor $D\colon\mathcal{I}^\mathsf{op}\times\mathcal{I}\to\mathcal{C}$ is the representing object of the functor
$$\mathrm{Wedges}_{(-)}(D)\colon\mathcal{C}^\mathsf{op}\to\mathsf{Sets}$$
with
$$\mathrm{Wedges}_{(-)}(D)\overset{\mathrm{def}}{=}\mathrm{ExtNat}(\overline{\Delta_{(-)}},\overline{D}),$$
where

- $\overline{D}\colon\mathsf{pt}\times\mathcal{I}^\mathsf{op}\times\mathcal{I}$ is the unique functor restricting to $D$ under the isomorphism $\mathsf{pt}\times\mathcal{I}^\mathsf{op}\times\mathcal{I}\cong\mathcal{I}^\mathsf{op}\times\mathcal{I}$ and similarly for $\overline{\Delta_{(-)}}$, and where
- We are now working with extranatural transformations.

That is, the object $\int_{A\in\mathcal{C}}D^A_A$ of $\mathcal{C}$ such that $$h_{\int_{A\in\mathcal{C}}D^A_A}\cong\mathrm{Wedges}_{(-)}(D).$$

## Recall: Weighted Limits

We can generalise limits by replacing $\Delta_{(-)}$ with an arbitrary functor $W\colon\mathcal{C}\to\mathsf{Sets}$. This leads to the notion of the **weighted limit** of $D\colon\mathcal{I}\to\mathcal{C}$ with respect to the **weight** $W$. This is the object $\lim_W(D)$ of $\mathcal{C}$ for which we have a natural isomorphism
$$h_{\lim_W(D)}(-)\cong\mathrm{Nat}(W,\mathrm{Hom}_\mathcal{C}(-,D)).$$

## Question: Weighted Ends

Just as with weighted limits, we may define the **weighted end** of a functor $D\colon\mathcal{I}^\mathsf{op}\times\mathcal{I}\to\mathcal{C}$ with respect to a **weight** $W\colon\mathcal{I}^\mathsf{op}\times\mathcal{I}\to\mathsf{Sets}$ as the object $\int_{A\in\mathcal{C}}^W D^A_A$ of $\mathcal{C}$ (if it exists) such that we have a natural isomorphism
$$h_{\int_{A\in\mathcal{C}}^W(D)}(-)\cong\mathrm{ExtNat}(\overline{W},\overline{\mathrm{Hom}_\mathcal{C}(-,D)}).$$
(Or rather that a certain natural transformation $W_*$ induced by $W$ is a natural isomorphism. Note that precomposing extranatural transformations with natural transformations gives back an extranatural transformation, so $\mathrm{ExtNat}(\cdots)$ is indeed a functor.)

Now (finally!) for the actual questions:

- This notion seems to be very natural. Has it been considered somewhere in the literature?
- Provided that $\mathcal{C}$ has cotensors, we may write any weighted limit on $\mathcal{C}$ as an end. Can we similarly express weighted ends in terms of ends or limits (possibly weighted)?
- Are there any natural occuring examples of this notion?
- Everything above can be categorified to the setting of bicategories (with pain). Is there anything remarkable about the resulting notion of a "weighted pseudo biend"?