§ Reduction to discrete category

• Let's take a functor category $[X, Y]$.
• Take a diagram $D: J \to [X, Y]$. What is the limit $\lim D: [X, Y]$?
• First, let's assume that $X$ has no arrows, or that we forget all the arrows of $X$ except the identity arrows. denote this forgotten/discrete category by $ob(X)$, whose objects are those of $X$, and morphisms are only identity morphisms.
• We can define the diagram $ob(D): J \to [ob(X), Y]$. Can we compute $\lim ob(D)$?
• A functor $ob(X) \to Y$ is the same as a tuple $Y^{ob(X)}$. See that $Y^{ob(X)}$ lives in CAT, since it is a category that is the $ob(X)$ copies of $Y$.

§ Formal proof by limits of product categories

• Now, the limit $ob(D)$ can be interpreted as a limit of $ob(D): J \to Y \times Y \times \cdots \times Y$.
• By the universal property of the product, limits over product categories can be computed pointwise . So if we have a diagram $E: K \to X \times Y$, then $l \equiv \lim E$ can be calculated by calculating $l_x \equiv \lim (\pi_1 \circ E : K \to X)$, then $l_y \equiv \lim (\pi_2 \circ E : K \to Y)$, and then setting $l \equiv (l_x, l_y) \in X \times Y$.
• Thus, we split the morphism $ob(D): J \to Y \times Y \times \cdots \times Y$ into the individial tuple components, which correspond to the images of $x \in ob(X)$ under $D$, 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 ob(X) to Y.
• Denote 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 Fp and Fq.
• 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 F to H and G to 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(D) in [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:

[Fp     Fq] |       | αp     αq v       v[Hp      Hq] ^       ^ βp      βq |       |[Gp      Gq]

• 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 [X, Y]

• We now believe that given $D: J \to [X, Y]$, we know that we can compute $ob(D): J \to [ob(X), Y]$ pointwise.
• Formally, we define $[\lim ob(D)](x)$ to be equal to $\lim (ev_x \circ D : J \to Y)$.
• We define the action of $\lim D$ (which is a functor from $X$ to $Y$) on objects of $X$ to be equal to the action of $\lim ob(D)$ on objects of $X$, which is given by the above equation.
• So what about the action of $\lim D$ on the morphisms of $X$? it's a functor from $X$ to $Y$, so it should send morphisms to morphisms!
• Now, let's suppose we have a morphism $x \xrightarrow{a} x'$ in $X$. How do we compute the the action of $D$ on the morphism $a$?
• Well, first off, what's $D(a)$ a morphism between? It must be between $D(x)$ and $D(x')$.
• What is $D(x)$? We know that $D(x) \equiv \lim (ev_x \circ D: J \to Y)$. Similarly, we know that $D(x') \equiv \lim ev_x' \circ D: J \to Y)$.