§ 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 .
- The category must be monoidal, so we can define composition as .
§ Weighted Limits via collages
1 be the terminal enriched category, having 1 object
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
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
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
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
- Terminal means what it usually does. A terminal weighted cone is a weighted limit.
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
- Let be a small category, which is the diagram.
- Suppose .
- See that cones of corresond to natural transformations for .
- See that the limit represents cones: , natural in
- Generalizing this to arbitrary category , we can write