## § 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 wefinish 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 controlthe first chapter (existential).
• The inverse limit is like writing a book from a very rough outline to a moredetailed outline. The first outline will be very vague, but it controls theentire narrative (universal). But this can be refined by the later draftswe perform, and can thus be "refined" / "cauchy sequence'd" into somethingfiner.

#### § 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 issome sort of universal quantification.