- more explicitly, for
`P : C -|-> D`

, define`Collage(P)`

as the category where`Obj(Collage(P)) = Obj(D) + Obj(C)`

,`Collage(P)(inl x, inl y) = D(x,y)`

,`Collage(P)(inr x, inr y) = C(x,y)`

,`Collage(P)(inl x, inr y) = P(x,y)`

,`Collage(P)(inr x, inl y) = 0`

- It is the categorification of a cograph. A graph is where we take the product
`A \times B`

and then take a subset of it where`f(x) = y`

(equalizer). - A cograph is where we take the union
`A \cup B`

and then impose a quotient`f(x) ~ y`

(coequalizer). - When we categorify this, we don't coequalize, but we setup arrows that capture the morphisms.

- In an enriched category, we replace hom sets by hom objects which live in some suitable category $V$.
- The category must be monoidal, so we can define composition as $\circ: hom(y, z) \otimes hom(x, y) \to hom(x, z)$.

- Let
`1`

be the terminal enriched category, having 1 object`*`

and`Hom(*,*) = I`

, and`I`

is the unit of the monoidal structure`(V, (x), I)`

of the enrichment. - A weighted cone over
`D : J -> C`

with weight`W : J -|-> 1`

(where`I`

is the terminal enriched category over`V`

), is a functor`G`

from the collage of`W: J -|-> 1`

to`C`

that agrees with`F`

on the copy of`J`

in the collage. So,`G: Col(W) -> C`

, or`G: J+* -> C`

where`G(J) = D`

. - Unravelling this, construct the category
`Col(W) = J+*`

with the morphisms in`J`

, morphism`I: * -> *`

, and a bunch of arrow`J -> *`

. So we are adding an "enriched point", with an arrow`I: * -> *`

. - What does a weighted cone
`G: Col(W) -> C`

have that doesn't just come from`F: J -> C`

? Well, it has an object`X`

(for apeX) to be the image of`(inr *): J+*`

, and it has the collage maps`W(inl j -> inr *) -> C(j -> X)`

for all`j`

in`Obj(J)`

, and these maps commute with the base maps of`F`

. So far, this looks like a cone. However, note that the collage maps are enriched maps! - The natural transformations can only choose to move where
`*`

goes, since that's the only freedom two functors`G, G:':J+* -> C`

have, since they must agree with`F`

on`J`

:`G(J) = G'(J) = F(J)`

. This is akin to moving the nadir, plus commutation conditions to ensure that this is indeed a cone morphism.

- Maps of these weighted cones are natural transformations that are identity on the copy of
`J`

- Terminal means what it usually does. A terminal weighted cone is a weighted limit.

`15:51 ` *C(X,F(j))
How does this look in our ordinary Set-enriched world? a `W`-weighted cone has its ape`X` and for each `j` in `J`
it has a `W(*,j)`-tuple of arrows `x_j,k : X -> F(j)` in `C` and for each `g : j -> j'` we have equations `x_j,k . F(g) = x_j',W(*,g)(k)`
15:57 both correct
15:57 wait, no
15:58 first correct
15:58 maps of weighted cones are natural transformations `eta : F => F' : Collage(W) -> C` that are identity on the copy of J in Collage(W)
16:04 in the `Set`-enriched world, a map of `W`-weighted cones is a map `f : X -> X'` in `C` and for each `j` in `Obj(J)` and `k` in `W(*,j)` we have equations `x_j,k = x'_j,k . f`
16:08 so you can take a simple example, the second power. For this example, `J = 1`, `W(*,*) = 2`, `F` picks out some object `c`, so each weighted cone consists of `X` and `x_*,0 : X -> c` and `x_*,1 : X -> c` and no equations
16:09 what does the terminal weighted cone look like in this example?

`nlab`

- Let $K$ be a small category, which is the diagram.
- Suppose $F: K \to \mathsf{Set}$.
- See that cones of $F$ corresond to natural transformations $[K, \mathsf{Set}](\Delta(p), F)$ for $p \in \mathsf{Set}$.
- See that the limit represents cones: $\mathsf{Set}(p, \texttt{Lim} F) \simeq [K, \mathsf{Set}](\Delta(p), F)$, natural in $p$
- Generalizing this to arbitrary category $C$, we can write