$P(q)^\infty \equiv \{ \texttt{exp}(2\pi k /q^n) : \forall n, k \in \mathbb N, ~ 0 \leq k \leq q^n \}$

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.

- Every subgroup of the prufer group is finite.
- By Lagrange,
`|prufer|/|subgroup| = |quotient|`

. But this gives us something like`infinite/finite = infinite`

.

```
... < 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.
`A*`

$L \equiv \{ a[:] \in \prod_i A_i : \texttt{proj}(\alpha \leftarrow \omega)(a[\omega]) = a[\alpha] ~ \forall \alpha \leq \omega \}$

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.

`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 - Having solutions to some equation in $\mathbb{Z}/7\mathbb{Z}$
- Finding a solution in $\mathbb{Z}/49\mathbb{Z}$ that restricts to the same solution in $\mathbb{Z}/7\mathbb{Z}$
- Keep going.

- in $\mathbb{Z}/7\mathbb{Z}$ we have that $2 \times 4 \equiv 1$.
- in $\mathbb{Z}/49\mathbb{Z}$ $2 \times 4 \equiv 8$.

$\{ (P_0, P_1, P_2, \dots) \in \prod_{i=0}^n \Pi_n : P_a = \texttt{proj}_{a \leftarrow z}(P_z) \forall a \leq z \}.$

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".
$\begin{aligned}
&(1) \in V_1 \\
&(1, 1) \in V_2 \\
&(1, 1, 1) \in V_3 \\
&(1, 1, 1, 1) \in V_4 \\
&\dots
\end{aligned}$

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?
- 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.

- 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.