Is there a discipline in Mathematics that studies information loss on the formal system level?
Mathematicians try to use hypothesis that are the weakest possible, meaning one goal of math is to limit the amount of information loss in theorems. However, it's not that big of a deal to use theorems that crush some minor statements even if that implies ignoring a huge amount of what the theorem tells us. So far as I understand it, maths is more about proving things in a certain environment (not quite sure what term to use here) and slowly but surely constructing theorems, thus the information is always kept in the hypothesis.
What I want to know however; is whether or not there is a discipline in mathematics that studies the amount and type of information that is lost when following a mathematical method. So for each step of some general object, for each step of the progress; we restrict the general case in order to arrive at a narrow conclusion, which is the strong theorem notation I am trying to seek. I wish to check how much is thrown away at each step; which should be captured by the restriction.
I have been advised by chat to link two pages in this question as they contain specific situations that the discipline should be able to describe:
On this page information is lost between the sets that functions are defined with, when the two are made into a composite function. Operation of function and relation with their domain and range
Here is a wiki page on the topic of sheafs. Secret mentioned this to me a discipline that seemed to describe what I was looking for but on the topic of topology only. https://en.wikipedia.org/wiki/Sheaf_(mathematics)
TheGreatDuck mentioned that I could perhaps say that it is information loss in general. He also stated that chances are, they'll just say "no it's not formally studied cause it's too broad"
My gut feeling agrees with this. However there may be a few cases in narrower disciplines when it is used, and they would be interesting to look up on if I was given a reference.