So I've been reading about derived categories recently (mostly via Hartshorne's *Residues and Duality* and some online notes), and while talking with some other people, I've realized that I'm finding it difficult to describe what a "triangle" is (as well as some other confusions, to be described below).

Let $\mathcal{A}$ be an abelian category, and let $K(\mathcal{A})$ and $D(\mathcal{A})$ be its homotopy category and derived categories respectively.

By definition, in either $K(\mathcal{A})$ or $D(\mathcal{A})$,

(1) A triangle is a diagram $X\rightarrow Y\rightarrow Z\rightarrow X[1]$ which is isomorphic to a diagram of the form $$X\stackrel{f}{\rightarrow}Y\rightarrow Cone(f)\rightarrow X[1]$$

This is the definition, though I don't really understand the motivation.

Somewhat more helpful for me, is the definition:

(2) A cohomological functor from a triangulated category $\mathcal{C}$ to an abelian category $\mathcal{A}$ is an additive functor which takes triangles to long exact sequences.

Since taking cohomology of a complex (in either $K(\mathcal{A})$ or $D(\mathcal{A})$) is a cohomological functor, this definition tells me that I should think of a triangle as being like "a short exact sequence" (in the sense that classically, taking cohomology of a short exact sequence results in a long exact sequence). This idea is also supported by the fact:

(3) If $0\rightarrow X^\bullet\rightarrow Y^\bullet\rightarrow Z^\bullet\rightarrow 0$ is an exact sequence of chain complexes, then there is a natural map $Z^\bullet\rightarrow X[1]^\bullet$ in the derived category $D(\mathcal{A})$ making $X^\bullet\rightarrow Y^\bullet\rightarrow Z^\bullet\rightarrow X[1]^\bullet$ into a triangle (in $D(\mathcal{A})$).

This leads to my first precise **question**: Can there exist triangles in $D(\mathcal{A})$ which don't come from exact sequences? If so, is there a characterization of them? Is (3) false in the homotopy category $K(\mathcal{A})$? (certainly the same proof doesn't work).

I sort of expect that the answers to the first and last questions above to both be "Yes", which makes the comparison between triangles and "exact sequences" a bit weird. Of course, $K(\mathcal{A})$ and $D(\mathcal{A})$ are almost never abelian categories, and so it's "weird" to talk about exact sequences there.

I suppose at a fundamental level, I find the "homotopy category" somewhat mysterious. I don't completely understand it's role in the construction of the derived category, since after all homotopy equivalences *are* quasi-isomorphisms. I also find it difficult to internalize this notion of "homotopic morphisms of chain complexes". To me, it's just a "technical trick" which allows one to do all this magic with mapping cones which allows $K(\mathcal{A})$ to be a triangulated category, whereas the normal abelian category of chain complexes is not. I can prove things with it, but whenever I do, I sort of feel unsettled - as if I'm playing with something 'magical' that could, at any moment, turn on me unexpectedly.

In addition to my specific questions above, I suppose I was hoping that someone would be able to articulate in a nice way how one should think of triangles, why this notion of a triangulated category is so successful, and hopefully alleviate some of my unsettlement regarding homotopy.