Short Exact Sequences as Fiber Bundles I am very much a visual thinker, and so to me, to feel that I have understood a concept, it is rather important for me to be able to "see a picture of it in my head". Now I recognize that for many concepts, this is bluntly impossible--I don't think that I ever will be able to think in 5-dimensional Euclidean space, for instance--but I want to at least be able to get an "approximate" or "cartoony" or "conceptually accurate" illustration in my mind.
So for instance, take a fiber bundle. It was introduced to me in one lecture as $(E,B,\pi,F)$: total space, base space, projection, and fiber, and that was it. As far as the lecturer was concerned, he had given a perfectly reasonable description of what a fiber bundle was. No need to dwell on it further than that. And sure, in a sense, I "knew" what a fiber bundle was after that lecture, but I feel that it was first when I got home later that day and googled and started seeing pictures of hairbrushes and Möbius stripes and other things that I actually knew what a fiber bundle was.
So, a topic that has long confused me are short exact sequences. When they were first introduced to me, the concept, though straightforward, seemed eminently arbitrary: the image of one map is the kernel of the next. Why is that interesting? Sure, from differential geometry and what little algebraic topology I knew at the time, I was aware of that such sequences of structures and maps between them frequently occurred in mathematics, but I could not see why they were interesting structures in and of themselves. What was the concept that they encoded?
I feel that my understanding is much better now, having had a look at some questions posted by other Stack Exchange users with the same stumbling block as me, such as What are exact sequences, metaphysically speaking? and Intuitive meaning of Exact Sequence. Nevertheless, I am still not 100% certain, and so I am posting this question in the same vein to see with more knowledgeable users if this understanding is correct:
A way I've heard short exact sequences explained by many people is that they are meant to give an illustration of structures that are "almost product spaces", in the sense that
$$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$
along with the trivial
$$0 \rightarrow A \rightarrow A \oplus C \rightarrow C \rightarrow 0$$
implies that $B$ is "almost" $A \oplus C$. This makes me wonder, is it fair to think of short exact sequences as "generalized fiber bundles", or fiber bundles where the "spaces" are groups, rings, modules, etc., akin to how the Möbius stripe is "almost" a cylinder?
Look forward to your responses!
 A: I will attempt to clarify fibre bundles. I think the reason you are finding them hard to grasp visually is because of the old duck and rabbit picture. In one way of looking at it, it is a duck, and the other a rabbit.
These two interpretations are whether you see a sample space (a set with elements used in statistics and probability) as a spectrum or as a neighbourhood. Each of these are topological invariants in an Hilbert space and a Banach space respectively.
If you are looking at a fibre bundle as a spectrum, then we call this a Class. 
Conversely, if we looking at one as a neighbourhood it is a a Character.
To cover them individually, a class is a phase space that is confined by curvature, such that the projection is affine and is unique to the embedding. This phase space is usually the solution to some differential equation, partial or no, that specifies the action or continuity of the partition of the topography. These are known as diffeomorphisms. In essenence, class numbers are the partitions of a boundary, which the 'taxi cab' number being the supremum of the modular form.
A character is an expanded null-space, in which the basis for the bundle resolves the functoid as a resultant. These are known as transforms. I find these tricky as they stay firmed rooted in algebra even though you are using harmonic analysis.
Class characters are combined in mathematics to create algebraic varieties, these are pretty null recursions you reference in your question. In higher dimensions these are called Chern-Calabi classes and in physics, particularly quantum mechanics, are known as revivals.
