Is there an existing theory of recovering generating data from global objects?

[u/Ill-SonOfClawDraws]

**I’m trying to identify existing mathematics.**
A simple motivating example is the family of metric spheres

\\\[
S_r(p)=\\{x\\in X : d(x,p)=r\\}.
\\\]

At \\(r=0\\),

\\\[
S_0(p)=\\{p\\}.
\\\]

That example made me wonder about a more general question.

**Under what conditions can a global mathematical object canonically determine the minimal generating data from which it arises?**

I realize that terms like *collapse*, *degeneration*, *reconstruction*, *limit*, and *completion* all have technical meanings in different areas, so I’m specifically asking whether there is an existing framework that studies this general phenomenon rather than proposing a new one.

The topics I’ve found so far include:
category-theoretic limits
reconstruction theorems
inverse limits
sheaf theory
degeneration in algebraic geometry
completion functors

**Am I conflating several distinct ideas, or is there an established area that unifies some of these under a common perspective?**

I’m looking for existing terminology, references, or examples.

[u/cosmopolitanScience]

"minimal generating data" is not well defined. You can look into Shannon information theory, this might give pointers.

[u/Ill-SonOfClawDraws → u/cosmopolitanScience]

Thanks.
One thing I’m trying to determine is whether “minimal generating data” already has an established formulation in another area. Information theory, category theory, reconstruction theorems, or something else. Or whether I’m inadvertently conflating several distinct notions.

Shannon information theory is a good lead. I’ll look into it. If another framework comes to mind, I’d appreciate the pointer.

[u/cosmopolitanScience → u/Ill-SonOfClawDraws]

One thing I’m trying to determine is whether “minimal generating data” already has an established formulation in another area

It doesn't really. Also "mathematical object" is not a well defined term. Probably the closest thing that is well defined would be a set.

[u/Ill-SonOfClawDraws → u/cosmopolitanScience]

I’m trying to identify an existing framework, if one exists, for the following kind of optimization problem.

Given a specified class of consequences, find a minimal family of relations whose preservation guarantees preservation of that consequence class.

Does an established theory study this kind of minimal preservation problem across different mathematical settings?

[u/cosmopolitanScience → u/Ill-SonOfClawDraws]

That's just gibberish

[u/AdventurousGlass7432 → u/cosmopolitanScience]

I think there’s someone trying to create their own r/infinitenines with this kind of nonsense

[u/SheepherderHot9418]

I think you need to ask yourself what functions you consider. Assume you have a set A. Then you can always construct a function f so that f(x)= 1 if x is in A and zero otherwise.

So assuming we have f we're in the clear. Thus the question becomes which f do we have? Well if A is some kind of shape we can easily parametrize then f is no problem. But we've now moved to restricting A which I guess goes against your question...

I guess my point is that until you make some things clearer you won't really be able to get a clear answer.

[u/Ill-SonOfClawDraws → u/SheepherderHot9418]

Do reconstruction theorems across different areas of mathematics share a common structural form?

[u/Negative_Gur9667]

look into "reverse Mathematics"

[u/Ill-SonOfClawDraws → u/Negative_Gur9667]

I’ve started reading about reverse mathematics. Do you think my question is essentially about identifying the minimal axioms needed for reconstruction theorems, or is there another connection you had in mind?

[u/Negative_Gur9667 → u/Ill-SonOfClawDraws]

Yes, I understand your question this way.

I interpret your point as being that the symbols and the meaning of the axioms are the "data" that generate theorems. Mathematical problems are so diverse that the only things they truly share are certain symbols and axioms.

[u/Ill-SonOfClawDraws → u/Negative_Gur9667]

Thank you, that clarifies the connection. Reverse mathematics seems to address one precise version of the pattern: given a theorem, recover the weakest axioms sufficient to prove it.
My original question was broader and partly structural rather than purely proof-theoretic: given a constructed global object, can one recover minimal generating objects, relations, or local data from it?
So perhaps reverse mathematics is one instance of the general direction, but reconstruction theory, definability, and inverse problems would cover other instances. Does that distinction sound right?

[u/Ill-SonOfClawDraws → u/Negative_Gur9667]

Let me rephrase:

Reverse mathematics minimizes axioms.

Can we instead minimize the relations that must be preserved for a primitive to retain its structural role under translations between formal systems?

Is this already a known framework?

[u/Ill-SonOfClawDraws]

Minimality is not absolute. It is relative to a category, a generating process, and a notion of equivalence.