`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".