§ Weighted limits via collages

§ Collage of a profunctor.

§ Quick intro to enriched (pro)functors.

§ Weighted Limits via collages

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