Another question came to my mind right after the answer to this one (Specific Question about $\omega_1$ hierarchies). Sorry if this is too many questions (I will delete this one if it doesn't turn out to be a good question).
Given a countable collection $C$ of subsets of $\mathbb{N}$, is there some informal "method"(or alternatively, are there standard examples) which extracts an increasing function (say $f:\mathbb{N} \rightarrow \mathbb{N}$) from $C$ such that for all sets in $A \in C$ we have $A \leq_T f$. What I mean is that how much difficult it can be (in general) to extract such a function. Given the answer to the linked question, the rough converse to this (extracting $C$ from a given function that grows faster than $f$) certainly doesn't seem possible in any fixed fashion.
But my question is that given $C$, is there some method (or at least, standard examples) that extracts the function $f$ in the above paragraph. For example, rather famously, if $C$ denotes halt computable sets we have a somewhat simple procedure to extract this function $f$.
But this question isn't about very small collections $C$, but potentially very broad classes (where $C$, for example, might also contain non-hyperarithmetic sets etc.). Or possibly various collections that possibly turn up in broader definitions of computability.
Or is it the case that extracting such a function is usually too cumbersome/unnatural and doesn't give us any more perspective than what directly what $C$ gives us (even for rather standard examples of $C$).
I understand that this is a broad question (the more examples the better), but also deliberately so.
It's not hard to do this, but as far as I know it's generally not useful - as you say, it doesn't tend to give any new information. That said, here's one approach which I'll break into steps. Below, "set" means "set of natural numbers" and "function" means "function from natural numbers to natural numbers."
First, note that any function can be turned into an increasing function in an information-preserving way: given $f:\omega\rightarrow\omega$, we let $f^{inc}$ be the function defined recursively by $$f^{inc}(0)=f(0),\quad f^{inc}(i+1)=f^{inc}(i)+f(i+1)+1$$ (the "$+1$" being used to handle the possibility that $f$ outputs $0$; if we dropped it, $f^{inc}$ would only be guaranteed to be nondecreasing). It's clear that $f^{inc}$ is increasing and computable from $f$, and given $f^{inc}$ we can compute $f$ recursively by $$f(0)=f^{inc}(0),\quad f(i+1)=f^{inc}(i+1)-f^{inc}(i)-1.$$
So it's enough to produce an arbitrary function computing each set in $C$ (note that we may assume for simplicity that $C$ is closed under $\le_T$, since each set only computes countably many sets).
But this is easy: given sets $C_i$ ($i\in\omega$), we just let $f(\langle i,j\rangle)=0$ if $j\not\in C_i$ and $f(\langle i,j\rangle)=1$ if $j\in C_i$, where "$\langle \cdot,\cdot\rangle$" is your favorite computable pairing operation. Clearly $f$ computes each $C_i$, and so we're done.
At this point it's worth mentioning an important side fact - that we can never exactly capture $C$ this way, except in trivial cases. Specifically:
This is a consequence of the Exact Pair Theorem, which is my favorite theorem in computability theory and ruins many lovely hopes and dreams. It's proved by forcing, of course, because forcing makes everything better.
So given $C$, we produce an $f$, but this $f$ will compute lots of things it "shouldn't." Intuitively, we can't collapse an unordered collection (as opposed to a list) of sets/functions into a single set/function and preserve the "information-theoretic picture," and this is a big part of why constructions such as the above simply aren't very useful.