§ Limits of a functor category are computed pointwise.
§ Reduction to discrete category
- Let's take a functor category .
- Take a diagram . What is the limit ?
- First, let's assume that has no arrows, or that we forget all the arrows of except the identity arrows. denote this forgotten/discrete category by , whose objects are those of , and morphisms are only identity morphisms.
- We can define the diagram . Can we compute ?
- A functor is the same as a tuple . See that lives in
CAT, since it is a category that is the copies of .
§ Formal proof by limits of product categories
- Now, the limit can be interpreted as a limit of .
- By the universal property of the product, limits over product categories can be computed pointwise . So if we have a diagram , then can be calculated by calculating , then , and then setting .
- Thus, we split the morphism into the individial tuple components, which correspond to the images of under , and we compute their limits. So we can compute this pointwise.
§ Draw the right diagram.
- Suppose we had
J = (f -a-> h <-b- g), and we had
ob(X) = (p q). We only have objects, no morphisms.
- Now, what is a diagram
ob(D): J -> [ob(X), Y]? For each of
f, g, h in
J, we must get a functor from
F = ob(D)(f),
G = ob(D)(g), and
H = ob(D)(h). Each of
F, G, H are functors
ob(X) -> Y.
- I'll write the functors by identifying them by their image. The image of
F is going to be
[Fp Fq]with no interesting morphisms between
- Now, that we've considered the action of
ob(D) on objects of
J, what about the arrows?
- The images of the arrows
f -a-> h and
h <-b- g are natural transformations from
H respectively. Denote these by
F =α>= H and
H <=β=G. So we have
ob(D)(a) = α,
ob(D)(b) = β.
- In total, the image of
[ob(X), Y] looks like this:
F =α=> H <=β= G
- If we expand out the functors by identifying them with the image, and write the natural transformations in terms of components, it looks like so:
- Really, the diagram consists of two parts which don't interact: the part about
p and the part about
q. So computing limits should be possible separately!
§ This extends to
- We now believe that given , we know that we can compute pointwise.
- Formally, we define to be equal to .
- We define the action of (which is a functor from to ) on objects of to be equal to the action of on objects of , which is given by the above equation.
- So what about the action of on the morphisms of ? it's a functor from to , so it should send morphisms to morphisms!
- Now, let's suppose we have a morphism in . How do we compute the the action of on the morphism ?
- Well, first off, what's a morphism between? It must be between and .
- What is ? We know that . Similarly, we know that .