## § 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.

#### § Prufer group

Here, the idea is to build a group consisting of all the $p^n$th roots of unity. We can directly """define""" the group as:
$P(q)^\infty \equiv \{ \texttt{exp}(2\pi k /q^n) : \forall n, k \in \mathbb N, ~ 0 \leq k \leq q^n \}$
That is, we take $q^1$th roots of unity, $q^2$th roots of unity, and so on for all $n \in \mathbb N$. To build this as a direct limit, we embed the group $Z/q^n Z$ in $Z/q^{n+1}Z$ by sending: the $q^n$ th roots of unity to $q^{n+1}$th roots of unity raised to the power $q$. An example works well here.
• To embed $Z/9Z$ in $Z/27Z$, we send:
• $2 \pi 1 /9$ to $2 \pi 1/9 \times (3/3) = 2 \pi 3 / 27$.
• $2 \pi 2 /9$ to $2 \pi 6/27$
• $2 \pi 3 /9$ to $2 \pi 9 / 27$
• $2 \pi k / 9$ to $2 \pi (3k)/27$
• This gives us a full embedding.
The direct limit of this gives us the prufer group. We can see that the prufer group is "different" from its components, since for one it has cardinality $\mathbb N$. For another, all subgroups of the prufer group are themselves infinite. The idea is to see that:
• Every subgroup of the prufer group is finite.
• By Lagrange, |prufer|/|subgroup| = |quotient|. But this gives us something like infinite/finite = infinite.
To see that every subgroup $H$ of the prufer group is finite, pick an element $o$ outside of the subgroup $H$. This element $o$ will belong to some $Z/q^kZ$ for some $k \in \mathbb Z$ (since the direct limit has an elements the union of all the original elements modulo some equivalence). If the subgroup $H$ does not have $o$ (and thus does not contain $Z/q^kZ$), then we claim that it cannot contain any of the larger $Z/q^{k+\delta}Z$. If it did contain the larger $Z/q^{k + \delta}$, then it would also contain $Z/q^k$ since we inject $Z/q^k$ into $Z/q^{k+\delta}$ when building the prufer group. Thus, at MAXIMUM, the subgroup $H$ can be $Z/q^{k-1}Z$, or smaller, which is finite in size. Pictorially:
...         < NOT in H
Z/q^{k+1}Z  < NOT IN H
Z/q^kZ      < NOT IN H
---------
...         < MAYBE IN H, FINITE
Z/q^2Z      < MAYBE IN H, FINITE
Z/qZ        < MAYBE IN H, FINITE

The finite union of finite pieces is finite. This $H$ is finite.

#### § 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. However, this example is useful as it lets us contrast the finite and infinite string case. Here, we see that in the final limit $A*$, we will have all strings of finite length. (In the infinite strings case, which is an inverse limit, we will have all strings of infinite length)

#### § Vector Spaces over $\mathbb R$

consider a sequence of vector spaces of dimension $n$: $V_1 \rightarrow V_2 \dots V_n$. Here, we can also find projection maps that allows us to go down from $V_n$ to $V_{n-1}$, and thus this has much the same flavour as that of finite strings. In the limiting object $V_\infty$, we get vectors that have a finite number of nonzero components. This is because any vector in $V_{\infty}$ must have come from some $V_N$ for some $N$. Here, it can have at most $N$ nonzero components. Further, on emedding, it's going to set all the other components to zero.

#### § 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".

#### § Vector Spaces

I can project back from the vector space $V_n$ to the vector space $V_{n-1}$. This is consistent, and I can keep doing this for all $n$. The thing that's interesting (and I believe this is true), is that the final object we get, $V^\omega$, can contain vectors that have an infinite number of non-zero components! This is because we can build the vectors:
\begin{aligned} &(1) \in V_1 \\ &(1, 1) \in V_2 \\ &(1, 1, 1) \in V_3 \\ &(1, 1, 1, 1) \in V_4 \\ &\dots \end{aligned}
Is there something here, about how when we build $V_\infty$, we build it as a direct limit. Then when we dualize it, all the arrows "flip", giving us $V^\omega$? This is why the dual space can be larger than the original space for infinite dimensional vector spaces?

#### § 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).

#### § Poetically, in terms of book-writing.

• The direct limit is like writing a book one chapter after another. Once we finish a chapter, we can't go back, the full book will contain the chapter, and what we write next must jive with the first chapter. But we only control the first chapter (existential).
• The inverse limit is like writing a book from a very rough outline to a more detailed outline. The first outline will be very vague, but it controls the entire narrative (universal). But this can be refined by the later drafts we perform, and can thus be "refined" / "cauchy sequence'd" into something finer.

#### § Differences

• The direct limit consists of taking unions, and we can assert that any element in $D_i$belongs in $\cup_i D_i$. So this lets us assert that $d_i \in D_i$ means that $d_i \in L$, or $\exists d_i \in L$, which gives us some sort of existential quantification.
• The inverses limit consists of taking $\prod_i D_i$. So given some element $d_i \in D_i$, we can say that elements in $L$ will be of the form $\{d_1\} \times D_2 \times D_3 \dots$. This lets us say $\forall d_1 \in D_1, \{d_1\} \times D_2 \dots \in L$. This is some sort of universal quantification.