One can think of the same mathematical object in many different ways. For example take $\mathbb{R}$. One can think of this as
(assume necessary hypotheses and so on)
As a group.
As a one dimensional subspace.
As an oriented line.
As a vector space.
As a one vector.
As a mathematically field.
As a metric space.
As a topological space.
As a fibre bundle on a base space.
As a geometrical line.
and so on.
Now there are many such examples for many other mathematical objects. It's annoying to me that there are so many different viewpoints. Is there a formalism that can incorporate all, or most of the different viewpoints (not only the viewpoints I listed here) in one framework? For example you might have a super abstract object, then you take something like a "limit" (not as in a series of course) and then you end up with $\mathbb{R}$ as interpretation as an oriented line, then you can take a different "limit" and you end up with the interpretation of $\mathbb{R}$ as a group.
Of course you don't have to take $\mathbb{R}$ as I did here. It's just an example of what I mean, and figuring out what is wrong about my examples is not the point of my question. Also you don't have to take my idea about a "limit" too serious, it's just an idea from me which originates as analogy from physics where in some limit everything becomes classical and simple (which might raise the question if there exists a physical framework which can do all these things).
You'd probably be interested in category theory, it contains lots of nice objects through a few different constructions and gives different objects which are "dual" to them in a precise sense as well as maps between objects which are "natural".
You'll find that a lot of things you've come across before are specific cases of categorical constructions.
These notes from H. Simmons are pretty good iirc, if you're interested;
https://web.archive.org/web/20180713101607/http://www.cs.man.ac.uk/~hsimmons/zCATS.pdf.