I've learned over the course of the last years that some mapping $\lambda$ denoted :
$$ \lambda : \mathbb{N} \rightarrow 2^{\mathbb{N}}.$$
essentially means that for every natural number, you assign (or map) some set of natural numbers to it. At multiple times, I've wondered if there's some meaning behind this notation (more precisely the $2^{\mathbb{N}}$ part) that I'm not understanding, as it does not seem as straightforward as say :
$$ f : n \rightarrow 2n$$
(also just noted as $f(n) = 2n$), where you clearly map any $n$ to $2n$. Is this $2^{\mathbb{N}}$ just some notation that one has to get used to, or does it make sense somehow to note it this way? If it does make sense, why the choice of exponentiation, and why the $2$?
$2^{\mathbb N}$ denotes the power set of $\mathbb N$, i.e. the set of all subsets of $\mathbb N$. This notation is used because, for a finite set $A$, the size of the power set $2^A$ of $A$ is $2^{|A|}$.
However, $\lambda:n\to 2^{\mathbb N}$ still seems like weird notation to me: the $n$ is an element of $\mathbb N$ (presumably) but $2^{\mathbb N}$ is a set. Should it maybe be $\lambda:\mathbb N\to 2^{\mathbb N}$?