## § Direct and Inverse limits

#### § Direct limit: definition

A direct limit consists of injections $A_1 \rightarrow A_2 \rightarrow \dots$. It leads to a limit object $L$, which as a set is equal to the union of all the $A_i$. It is equipped with an equivalence relation. We can push data "towards" the limit object, hence it's a "direct" limit. So each element in $A_i$ has an equivalence class representative in $L$.

#### § $S_n$

We can inject the symmetric groups $S_1 \rightarrow S_2 \rightarrow \dots$. However, we cannot project back some permutation of $S_2$ (say) to $S_1$: if I have $(2, 1)$ (swap 2 and 1), then I can't project this back into $S_1$. This is prototypical; in general, we will only have injections into the limit, not projections out of the limit.

#### § Stalks

Given a topological space $(X, T)$ and functions to the reals on open sets $F \equiv \{ U \rightarrow \R \}$, we define the restricted function spaces $F|_U \equiv \{ F_U : U \rightarrow \mathbb R : f \in F \}$. Given two open sets $U \subseteq W$, we can restrict functions on $W$ (a larger set) to functions on $U$ (a smaller set). So we get maps $F|_W \rightarrow F|_U$. So given a function on a larger set $W$, we can inject into a smaller set $U$. But given a function on a smaller set, it's impossible to uniquely extend the function back into a larger set. These maps really are "one way". The reason it's a union of all functions is because we want to "identify" equivalent functions. We don't want to "take the product" of all germs of functions; We want to "take the union under equivalence".

#### § Finite strings / A*

Given an alphabet set $A$, we can construct a finite limit of strings of length $0$, strings of length $1$, and so on for strings of any given length $n \in \mathbb N$. Here, the "problem" is that we can also find projection maps that allow us to "chop off" a given string, which makes this example not-so-great.

#### § Categorically

Categorically speaking, this is like some sort of union / sum (coproduct). This, cateogrically speaking, a direct limit is a colimit.

#### § Inverse limit: definition

An inverse limit consists of projections $A_1 \leftarrow A_2 \leftarrow \dots$. It leads to a limit object $L$, which as a set is equal to a subset of the product of all the $A_i$, where we only allow elements that "agree downwards" .Formally, we write this as:
$L \equiv \{ a[:] \in \prod_i A_i : \texttt{proj}(\alpha \leftarrow \omega)(a[\omega]) = a[\alpha] ~ \forall \alpha \leq \omega \}$
So from each element in $L$, we get the projection maps that give us the component $a[\alpha]$.
These 'feel like' cauchy sequences, where we are refining information at each step to get to the final object.

#### § infinite strings

We can consider the set of infinite strings. Given an infinite string, we can always find a finite prefix as a projection. However, it is impossible to canonically inject a finite prefix of a string into an infinite string! Given the finite string xxx, how do we make it into an infinite string? do we choose xxxa*, xxxb*, xxxc*, and so on? There's no canonical choice! Hence, we only have projections, but no injections.

Consider the 7-adics written as infinite strings of digits in $\{0, 1, \dots, 6\}$. Formally, we start by:
1. Having solutions to some equation in $\mathbb{Z}/7\mathbb{Z}$
2. Finding a solution in $\mathbb{Z}/49\mathbb{Z}$ that restricts to the same solution in$\mathbb{Z}/7\mathbb{Z}$
3. Keep going.
The point is that we define the $7$-adics by projecting back solutions from $\mathbb{Z}/49\mathbb{Z}$. It's impossible to correctly embed $\mathbb{Z}/7\mathbb{Z}$ into $\mathbb{Z}/49\mathbb{Z}$: The naive map that sends the "digit i" to the "digit i" fails, because:
• in $\mathbb{Z}/7\mathbb{Z}$ we have that $2 \times 4 \equiv 1$.
• in $\mathbb{Z}/49\mathbb{Z}$ $2 \times 4 \equiv 8$.
So $\phi(2) \times \phi(7) \neq \phi(2 \times 7) = \phi(4)$. Hece, we don't have injections, we only have projections.

#### § Partitions

Let $S$ be some infinite set. Let $\{ \Pi_n \}$ be a sequence of partitions such that $\Pi_{n+1}$ is finer than $\Pi_n$. That is, every element of $\Pi_n$ is the union of some elements of $\Pi_{n+1}$. Now, given a finer partition, we can clearly "coarsen" it as desired, by mapping a cell in the "finer space" to the cell containing it in the "coarser space". The reverse has no canonical way of being performed; Once again, we only have projections, we have no injections. The inverse limit is:
$\{ (P_0, P_1, P_2, \dots) \in \prod_{i=0}^n \Pi_n : P_a = \texttt{proj}_{a \leftarrow z}(P_z) \forall a \leq z \}.$
But we only care about "adjacent consistency", since that generates the other consistency conditions; So we are left with:
$\{ (P_0, P_1, P_2, \dots) \in \prod_{i=0}^n \Pi_n : P_a = \texttt{proj}_{a \leftarrow b}(P_b) \forall a +1 = b \}.$
But unravelling the definition of $\texttt{proj}$, we get:
$\{ (P_0, P_1, P_2, \dots) \in \prod_{i=0}^n \Pi_n : P_a \supseteq (P_b) \forall a +1 = b \}.$
So the inverse limit is the "path" in the "tree of partitions".

#### § Categorically

Categorically speaking, this is like some sort of product along with equating elements. This, cateogrically speaking, a inverse limit is a limit (recall that categorical limits exist iff products and equalizers exist).