§ Weighted limits via collages

§ Collage of a profunctor.

• 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.

§ Quick intro to enriched (pro)functors.

• 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 $\circ: hom(y, z) \otimes hom(x, y) \to hom(x, z)$.

§ Weighted Limits via collages

• 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?

§ Weighted limit via 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

• Ref: nlab