What are exact sequences, metaphysically speaking?

Question

Why is it natural or useful to organize objects (of some appropiate category) into exact sequences? Exact sequences are ubiquitous - and I've encountered them enough to know that they can provide a very useful and efficient framework to work within. However, I have no idea what this framework truly is or why it is effective.

So, my questions are:

1.) what makes exact sequences natural objects to deal with?

2.) what do they encode, generally speaking? Or if you are unable to think of a satisfactory answer in general, what are some specific examples of exact sequences encoding some desirable property?

Please set me straight! It seems like all of the references that I've come across only encyclopedically develop the idea of an exact sequence, sparing the reader of any qualification or exposition. Thank you.

Note: I had surely expected this to be a question already asked here or on mathoverflow... But I could not find anything like it. My apologies if this question has already been addressed.

Answer 1

This was too long to put as a comment, I apologize if it doesn't help.

I don't know how totally accurate this is, but I like to think of (short) exact sequences as being algebraified versions of fiber bundles. Thus, putting $X$ in a short exact sequence $0\to Y\to X\to Z\to0$ indicates to me that $X$ is put together in some way from $Y$ and $Z$, and in such a way that, in a perfect world where everything is nice, is just the product of $Y$ and $Z$. Therefore, $X$ is some kind of "twisted product" of $Y$ and $Z$.

Thus, any time we are able to put $X$ into an exact sequene we should (in spirit) be able to tell properties of $X$ from properties of $Y$ and $Z$.

For example, knowing that $B$ is an abelian groups such that

$$0\to A\to B\to C\to 0$$

is a SES for $B,C$ also abelian groups tells me that $\text{rank}(B)=\text{rank}(A)+\text{rank}(C)$ (or more generally this works nicely for modules over PIDs).

The reason that SESs are such a convenient framework to deal with the notion of "put-togetheredness" is that we live in a fundamentally arrow obsessed world. Things phrased entirely in terms of arrows make us happy, because they are often easy to deal with.

Answer 2

One algebraic answer is that exact sequences are a natural abstraction of the notion of generators and relations. That is, let $R$ be a ring and $M$ a left $R$-module with generating set $S$. Then there is a canonical surjection $$R^S \xrightarrow{f} M \to 0.$$

The kernel of this surjection describes all the possible relations in $S$ and gives rise to a short exact sequence $$0 \to \text{ker}(f) \to R^S \xrightarrow{f} M \to 0.$$

If $R$ is a PID, then $\text{ker}(f)$ is free, so picking a basis for $\text{ker}(f)$ gives an irredundant set of relations among the generators. However, if $\text{ker}(f)$ is not free, then picking a defining set of relations $T$ (that is, a generating set in $\text{ker}(f)$) instead gives rise to an exact sequence $$0 \to \text{ker}(g) \to R^T \xrightarrow{g} R^S \xrightarrow{f} M \to 0.$$

If $\text{ker}(g)$ is not free, then... and so on. From this perspective we are thinking of exact sequences as resolutions.