An early edition of Lang's algebra textbook gives the famous exercise to
Take any book on homological algebra, and prove all the theorems without looking at the proofs given in that book.
Here are some of theorems you would find in such a book: the snake lemma, the five lemma, the nine lemma, the many unnamed lemmas for constructing long exact sequences in algebraic topology (e.g. Mayer-Vietoris). The only way I know to prove these theorems, and which I have read in homological algebra books, is by a diagram chase.
I have nothing against diagram chasing -- it's rather fun once you figure out how it works. But it also feels rather mechanical and devoid of insight.
- Is there any systematic way to think about/prove/remember theorems in homological algebra like the ones that I cited without having to resort to diagram chasing?
- Are the theorems of the type I mentioned small enough in number to be subsumed by some general theory which explains them?
- Is homological algebra anything more than a mindless subject used for its applications in other areas of math?
A satisfactory answer would allow us to (for instance) look at the hypotheses of the five lemma and then see immediately that the vertical arrow is an isomorphism.
Note: I am aware of the formalism of abelian categories, though not yet a competent user. It seems to me that abelian categories generalize the setting of homological algebra without clarifying its content. For instance, it seems like most people, when pressed to prove theorems of the type I mentioned in an abelian category, use Mitchell's Embedding Theorem to justify performing a diagram chase.