In a first course in homological algebra, the lecturer introduced derived functors as universal $\delta$-functors, whose universal property is splicing short exact sequences into long ones.
It so happens that I have read about the elegant homotopical approach to derived functors - as Kan extensions along the localization (into to the homotopy category), which offer the best homotopical (preserving weak equivalences) approximation to a given functor. This is in the setting of homotopical categories - categories with a class of arrows satisfying the 2-out-6 property.
Two things bug me especially:
- What is the homotopical significance of splicing short exact sequences to long exact sequences? Would it be reasonable to say the equivalence of the two definitions in abelian categories means preserving weak equivalences and splicing are different viewpoints of the same thing?
Where can I find actual rigorous proof that in the abelian setting, universal $\delta$-functors are Kan extensions along localizations? How can I rigorously formulate the equivalence of the notions of splicing and preserving weak equivalences?
Update: I have crossposted the part not addressed by Qiaochu Yuan's answer to MO. Since I'm really asking two separate questions, I figured this makes sense.
The homotopical significance of long exact sequences has to do with fiber sequences and their categorical duals, cofiber sequences. This is a long story and I don't know where it's told well and for students but the prototypical example for spaces is the following. If $f : E \to B$ is a map of spaces $e \in E, b \in B$ are basepoints such that $f(e) = b$, then the induced fiber sequence looks like
$$\dots \to \Omega F \to \Omega E \to \Omega B \to F \to E \to B$$
where $\Omega$ denotes the based loop space and $F$ is the homotopy fiber of $f$ at $b$. Taking $\pi_0$ of this fiber sequence gives the long exact sequence in homotopy. Dually, taking cofiber sequences of spaces gives the long exact sequences in homology and cohomology.
For chain complexes you know the homotopy cofiber as the mapping cone, and repeatedly taking mapping cones is one way to derive the long exact sequence in homology coming from a short exact sequence of chain complexes.
In general, taking fibers and cofibers are two of the simplest and most important examples of homotopy limits and colimits. From an unnecessarily modern point of view, homological algebra is a special case of the study of stable $\infty$-categories, and all of the "reasonable" functors between stable $\infty$-categories are exact: they preserve finite homotopy limits and colimits, and in particular preserve fibers and cofibers. The process of taking derived functors is the process of figuring out how to turn a functor between, say, abelian categories into a functor between stable $\infty$-categories, and turning short exact sequences into long exact sequences (which is a structure) is a reflection of the higher-categorical phenomenon of preserving fibers and cofibers (which is a property).