The Stacks project

Remark 13.4.4. Let $\mathcal{D}$ be an additive category with translation functors $[n]$ as in Definition 13.3.1. Let us call a triangle $(X, Y, Z, f, g, h)$ special1 if for every object $W$ of $\mathcal{D}$ the long sequence of abelian groups

\[ \ldots \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(W, X) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(W, Y) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(W, Z) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(W, X[1]) \to \ldots \]

is exact. The proof of Lemma 13.4.3 shows that if

\[ (a, b, c) : (X, Y, Z, f, g, h) \to (X', Y', Z', f', g', h') \]

is a morphism of special triangles and if two among $a, b, c$ are isomorphisms so is the third. There is a dual statement for co-special triangles, i.e., triangles which turn into long exact sequences on applying the functor $\mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(-, W)$. Thus distinguished triangles are special and co-special, but in general there are many more (co-)special triangles, than there are distinguished triangles.

[1] This is nonstandard notation.

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